Time Series Project: Summer 2023
Robert (Reuven) Derner
2023-08-05
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library(tidyr)Background
Our client Annaly Capital Management, Inc. is one of the largest mortgage real estate investment trusts. It is organized in Maryland with its principal office in New York City. The company borrows money, primarily via short term repurchase agreements, and reinvests the proceeds in asset-backed securities. CEO: David L. Finkelstein (Mar 2020–) has hired our firm to examine its past historical data and perform several time series forecasts to predict where the stock will open in the future based on current volumes. The data is contains the open, close, high, low price as well as the total volume recorded at every seven day period between the period 2013 - 2018.
We were asked for supplemental analysis coming from the consumer price index and mortgage rates for the period examined to see if these supplemental explanatory variable will have an effect on modeling.
Load in the data from github
df = read.csv("https://raw.githubusercontent.com/ReuvenDerner/MSDS-6373-Time-Series/master/Unit%205/NLY.csv")
cpi = read.csv("https://raw.githubusercontent.com/ReuvenDerner/TimeSeriesProject2023/main/CPI_CSV_2013_2018.xlsx%20-%20Sheet1.csv")
mort = read.csv("https://raw.githubusercontent.com/ReuvenDerner/TimeSeriesProject2023/main/MORTGAGE30US_2013_2018.csv")# take a sample of 15 from the dataframe
nyse_sample = sample_n(df, 5)
knitr::kable(nyse_sample, "html")| Date | Open | High | Low | Close | Adj.Close | Volume |
|---|---|---|---|---|---|---|
| 10/19/15 | 10.14 | 10.28 | 10.11 | 10.18 | 7.296029 | 27311100 |
| 3/27/17 | 11.14 | 11.37 | 11.02 | 11.11 | 9.213124 | 41196600 |
| 9/7/15 | 10.2 | 10.25 | 9.92 | 10.09 | 7.023324 | 28987400 |
| 4/11/16 | 10.16 | 10.45 | 10.16 | 10.41 | 7.926142 | 67314400 |
| 5/2/16 | 10.43 | 10.85 | 10.25 | 10.85 | 8.261157 | 48933200 |
cpi_sample = sample_n(cpi, 5)
knitr::kable(cpi_sample, "html")| Year | Month | CPI_Score |
|---|---|---|
| 2018 | Jan | 247.867 |
| 2016 | Mar | 238.132 |
| 2018 | Aug | 252.146 |
| 2018 | May | 251.588 |
| 2017 | Jun | 244.955 |
mort_sample = sample_n(mort, 5)
knitr::kable(mort_sample, "html")| DATE | MORTGAGE30US |
|---|---|
| 2014-07-17 | 4.13 |
| 2017-09-28 | 3.83 |
| 2018-08-09 | 4.59 |
| 2016-05-05 | 3.61 |
| 2018-11-15 | 4.94 |
The data has no missing values and no imputation is necessary, we can proceed with our analysis.
#reassign the dataframe
#df = TimeSeriesProject2018_2020
# Address the missing values in each column (NA as well as empty strings).
missing_df = as.data.frame(sapply(df, function(x) sum(is.na(x))))
colnames(missing_df) = c("variable missing")
knitr::kable(missing_df, "html")| variable missing | |
|---|---|
| Date | 0 |
| Open | 0 |
| High | 0 |
| Low | 0 |
| Close | 0 |
| Adj.Close | 0 |
| Volume | 0 |
empty_string_df = as.data.frame(sapply(df, function(x) sum(x == "")))
colnames(empty_string_df) = c("variable empty")
knitr::kable(empty_string_df, "html")| variable empty | |
|---|---|
| Date | 0 |
| Open | 0 |
| High | 0 |
| Low | 0 |
| Close | 0 |
| Adj.Close | 0 |
| Volume | 0 |
Generate summary statistics
# Generate summary statistics
summary(df)## Date Open High Low
## Length:263 Length:263 Length:263 Length:263
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
## Close Adj.Close Volume
## Length:263 Length:263 Length:263
## Class :character Class :character Class :character
## Mode :character Mode :character Mode :charactersummary(cpi)## Year Month CPI_Score
## Min. :2013 Length:72 Min. :230.3
## 1st Qu.:2014 Class :character 1st Qu.:236.3
## Median :2016 Mode :character Median :238.3
## Mean :2016 Mean :240.6
## 3rd Qu.:2017 3rd Qu.:244.8
## Max. :2018 Max. :252.9summary(mort)## DATE MORTGAGE30US
## Length:263 Min. :3.410
## Class :character 1st Qu.:3.815
## Mode :character Median :3.990
## Mean :4.039
## 3rd Qu.:4.280
## Max. :4.940We need to reclassify the data as numeric volumes, in addition we will have to adjust our input data from mortgage & consumer price index rates as additional columns in the data frame.
# Convert the Date column with the correct format
df$Date <- as.Date(df$Date, format = "%m/%d/%y")
# Convert columns to appropriate data types
df <- df %>%
mutate(Date = as.Date(Date), # Convert Date column to Date type
Open = as.numeric(Open), # Convert Open column to numeric type
High = as.numeric(High), # Convert High column to numeric type
Low = as.numeric(Low), # Convert Low column to numeric type
Close = as.numeric(Close), # Convert Close column to numeric type
Adj.Close = as.numeric(Adj.Close), # Convert Adj Close column to numeric type
Volume = as.integer(Volume)) # Convert Volume column to integer type## Warning: There were 6 warnings in `mutate()`.
## The first warning was:
## ℹ In argument: `Open = as.numeric(Open)`.
## Caused by warning:
## ! NAs introduced by coercion
## ℹ Run `dplyr::last_dplyr_warnings()` to see the 5 remaining warnings.summary(df)## Date Open High Low
## Min. :2013-11-18 Min. : 8.97 Min. : 9.04 Min. : 8.25
## 1st Qu.:2015-02-19 1st Qu.:10.21 1st Qu.:10.36 1st Qu.:10.03
## Median :2016-05-23 Median :10.59 Median :10.74 Median :10.40
## Mean :2016-05-22 Mean :10.75 Mean :10.91 Mean :10.57
## 3rd Qu.:2017-08-24 3rd Qu.:11.31 3rd Qu.:11.49 3rd Qu.:11.15
## Max. :2018-11-23 Max. :12.61 Max. :12.73 Max. :12.32
## NA's :1 NA's :1 NA's :1
## Close Adj.Close Volume
## Min. : 8.96 Min. : 5.572 Min. : 4209958
## 1st Qu.:10.21 1st Qu.: 7.000 1st Qu.: 33175550
## Median :10.57 Median : 8.060 Median : 40010450
## Mean :10.75 Mean : 8.291 Mean : 43003015
## 3rd Qu.:11.31 3rd Qu.: 9.854 3rd Qu.: 49882100
## Max. :12.60 Max. :10.994 Max. :197494800
## NA's :1 NA's :1 NA's :1NA were introduced by the mutation of the data, we will just remove the record where that occurs
df.adj = na.omit(df)Next lets examine CPI and MORT rates
cpi = read.csv("https://raw.githubusercontent.com/ReuvenDerner/TimeSeriesProject2023/main/CPI_CSV_2013_2018.xlsx%20-%20Sheet1.csv")
# Step 1: Create a new "Date" column in the "cpi" dataset by combining "Year" and "Month"
cpi$Date <- as.Date(paste(cpi$Year, cpi$Month, "01", sep = "-"), format = "%Y-%b-%d")
# Step 2: Select only the "Year", "Month", and "CPI_Score" columns from the "cpi" dataset
cpi <- cpi %>%
select(Year, Month, CPI_Score)
# Step 3: Extract "Year" and "Month" from the "df" Date column
df$Year <- format(df$Date, "%Y")
df$Month <- format(df$Date, "%b")
# Step 3: Convert "Year" column in "df" to integer
df$Year <- as.integer(df$Year)
# Step 4: Merge "df" and "cpi" datasets based on "Year" and "Month"
final_data <- df %>%
left_join(cpi, by = c("Year", "Month"))
# Step 5: Forward fill (carry the last observation forward) the "CPI_Score" values to fill missing values
final_data <- final_data %>%
group_by(Year) %>%
tidyr::fill(CPI_Score, .direction = "downup") %>%
ungroup()
#display final column to see output
head(final_data,10)## # A tibble: 10 × 10
## Date Open High Low Close Adj.Close Volume Year Month CPI_Score
## <date> <dbl> <dbl> <dbl> <dbl> <dbl> <int> <int> <chr> <dbl>
## 1 2013-11-18 NA NA NA NA NA NA 2013 Nov 233.
## 2 2013-11-25 10.2 10.3 9.86 10.2 5.81 77369900 2013 Nov 233.
## 3 2013-12-02 10.2 10.2 9.66 9.74 5.57 73844200 2013 Dec 233.
## 4 2013-12-09 9.7 10.3 9.69 9.94 5.69 65017500 2013 Dec 233.
## 5 2013-12-16 9.94 10.4 9.79 10.1 5.78 108105500 2013 Dec 233.
## 6 2013-12-23 10.1 10.3 9.81 9.83 5.62 52552600 2013 Dec 233.
## 7 2013-12-30 9.82 10.1 9.72 10 5.90 65043600 2013 Dec 233.
## 8 2014-01-06 10.0 10.3 9.97 10.2 6.04 63756100 2014 Jan 234.
## 9 2014-01-13 10.2 10.4 10.1 10.2 6.01 42784400 2014 Jan 234.
## 10 2014-01-20 10.2 10.7 10.2 10.5 6.21 42342100 2014 Jan 234.final_data2 = final_data
# Step 1: Convert the "DATE" column in mort to Date type
mort$DATE <- as.Date(mort$DATE)
# Step 2: Use approx() to find the closest "MORTGAGE30US" rate for each DATE in final_data
nearest_mortgage <- approx(mort$DATE, mort$MORTGAGE30US, final_data2$Date)$y
# Step 3: Add the "MORTGAGE30US" rate to the final_data dataset
final_data2$MORTGAGE30US <- nearest_mortgage
# Now, "final_data" contains the combined dataset with "CPI_Score" and "MORTGAGE30US" rates for each date.
head(final_data2)## # A tibble: 6 × 11
## Date Open High Low Close Adj.Close Volume Year Month CPI_Score
## <date> <dbl> <dbl> <dbl> <dbl> <dbl> <int> <int> <chr> <dbl>
## 1 2013-11-18 NA NA NA NA NA NA 2013 Nov 233.
## 2 2013-11-25 10.2 10.3 9.86 10.2 5.81 77369900 2013 Nov 233.
## 3 2013-12-02 10.2 10.2 9.66 9.74 5.57 73844200 2013 Dec 233.
## 4 2013-12-09 9.7 10.3 9.69 9.94 5.69 65017500 2013 Dec 233.
## 5 2013-12-16 9.94 10.4 9.79 10.1 5.78 108105500 2013 Dec 233.
## 6 2013-12-23 10.1 10.3 9.81 9.83 5.62 52552600 2013 Dec 233.
## # ℹ 1 more variable: MORTGAGE30US <dbl># Remove the columns "Year", and "Month" from the "final_data" dataframe
final_data2 <- final_data2 %>%
select(-Year, -Month)
#rename dataframe for use in modeling
finaldata <- na.omit(final_data2)There was two data points that had missing data contained within them. We removed these records from the data set to be used in modeling.
Examine any correlations that may exist
ggpairs(df.adj)
ggpairs(finaldata)
The adj.Close & Open Price is significantly correlated to all other variables in the dataset, particular with one another at 0.91 positive correlation.
The CPI_Score does have a very strong correlation to Date (time) and Adj.Close price. This makes sense given that the CPI score is regulated and reevaluated by the FED every month. In addition the overall average mortgage rate has a moderate positive correlation to CPI Score, as rates tend to rise, the overall consumer price also rises.
Check some data based on our response variable “Open”
## $xbar
## [1] 10.74882
##
## $autplt
## [1] 1.0000000 0.9558782 0.9140551 0.8798739 0.8356028 0.7947093 0.7579893
## [8] 0.7244069 0.6985809 0.6709464 0.6492885 0.6188956 0.5877319 0.5641738
## [15] 0.5247691 0.4840137 0.4481811 0.4072040 0.3655794 0.3241797 0.2839057
## [22] 0.2530625 0.2206851 0.2007074 0.1831944 0.1594867
##
## $freq
## [1] 0.003816794 0.007633588 0.011450382 0.015267176 0.019083969 0.022900763
## [7] 0.026717557 0.030534351 0.034351145 0.038167939 0.041984733 0.045801527
## [13] 0.049618321 0.053435115 0.057251908 0.061068702 0.064885496 0.068702290
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## [67] 0.255725191 0.259541985 0.263358779 0.267175573 0.270992366 0.274809160
## [73] 0.278625954 0.282442748 0.286259542 0.290076336 0.293893130 0.297709924
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## [127] 0.484732824 0.488549618 0.492366412 0.496183206 0.500000000
##
## $dbz
## [1] 12.3443460 12.1228799 11.7532303 11.2347782 10.5671017 9.7505942
## [7] 8.7875642 7.6841519 6.4535662 5.1212070 3.7316979 2.3555051
## [13] 1.0875438 0.0259685 -0.7692079 -1.3080599 -1.6600405 -1.9129555
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## [91] -15.3458703 -15.1992467 -14.9607755 -14.6658854 -14.3556431 -14.0692508
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## [127] -17.3715141 -17.4311300 -17.4934183 -17.5398684 -17.5570033
## $freq
## [1] 0.003816794 0.007633588 0.011450382 0.015267176 0.019083969 0.022900763
## [7] 0.026717557 0.030534351 0.034351145 0.038167939 0.041984733 0.045801527
## [13] 0.049618321 0.053435115 0.057251908 0.061068702 0.064885496 0.068702290
## [19] 0.072519084 0.076335878 0.080152672 0.083969466 0.087786260 0.091603053
## [25] 0.095419847 0.099236641 0.103053435 0.106870229 0.110687023 0.114503817
## [31] 0.118320611 0.122137405 0.125954198 0.129770992 0.133587786 0.137404580
## [37] 0.141221374 0.145038168 0.148854962 0.152671756 0.156488550 0.160305344
## [43] 0.164122137 0.167938931 0.171755725 0.175572519 0.179389313 0.183206107
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## [67] 0.255725191 0.259541985 0.263358779 0.267175573 0.270992366 0.274809160
## [73] 0.278625954 0.282442748 0.286259542 0.290076336 0.293893130 0.297709924
## [79] 0.301526718 0.305343511 0.309160305 0.312977099 0.316793893 0.320610687
## [85] 0.324427481 0.328244275 0.332061069 0.335877863 0.339694656 0.343511450
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## [97] 0.370229008 0.374045802 0.377862595 0.381679389 0.385496183 0.389312977
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## [115] 0.438931298 0.442748092 0.446564885 0.450381679 0.454198473 0.458015267
## [121] 0.461832061 0.465648855 0.469465649 0.473282443 0.477099237 0.480916031
## [127] 0.484732824 0.488549618 0.492366412 0.496183206 0.500000000
##
## $pzgram
## [1] 12.3443460 12.1228799 11.7532303 11.2347782 10.5671017 9.7505942
## [7] 8.7875642 7.6841519 6.4535662 5.1212070 3.7316979 2.3555051
## [13] 1.0875438 0.0259685 -0.7692079 -1.3080599 -1.6600405 -1.9129555
## [19] -2.1400160 -2.3896794 -2.6893036 -3.0518240 -3.4806765 -3.9722487
## [25] -4.5166737 -5.0981771 -5.6962174 -6.2883607 -6.8549197 -7.3838797
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## [43] -11.3872726 -11.5557692 -11.7445346 -11.9411914 -12.1261441 -12.2778885
## [49] -12.3791413 -12.4217684 -12.4085394 -12.3512550 -12.2667084 -12.1726483
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## [61] -12.2682448 -12.4624343 -12.6998127 -12.9694599 -13.2541297 -13.5299047
## [67] -13.7675699 -13.9367607 -14.0129350 -13.9850215 -13.8598244 -13.6604049
## [73] -13.4194473 -13.1714081 -12.9467196 -12.7690074 -12.6545893 -12.6131227
## [79] -12.6485770 -12.7601240 -12.9428284 -13.1881519 -13.4843191 -13.8165558
## [85] -14.1671642 -14.5154049 -14.8373456 -15.1062620 -15.2946569 -15.3788765
## [91] -15.3458703 -15.1992467 -14.9607755 -14.6658854 -14.3556431 -14.0692508
## [97] -13.8394649 -13.6909797 -13.6406322 -13.6982675 -13.8675239 -14.1461975
## [103] -14.5261126 -14.9926349 -15.5242113 -16.0926253 -16.6648862 -17.2073897
## [109] -17.6917666 -18.1000522 -18.4261423 -18.6724033 -18.8435710 -18.9417729
## [115] -18.9654783 -18.9126401 -18.7858615 -18.5962859 -18.3638336 -18.1138551
## [121] -17.8722552 -17.6612698 -17.4968983 -17.3878545 -17.3354561 -17.3339992
## [127] -17.3715141 -17.4311300 -17.4934183 -17.5398684 -17.5570033
## $xbar
## [1] 10.75188
##
## $autplt
## [1] 1.0000000 0.9558761 0.9140608 0.8783026 0.8343831 0.7937568 0.7564875
## [8] 0.7238239 0.6984386 0.6712387 0.6491151 0.6204025 0.5894214 0.5655573
## [15] 0.5263501 0.4840331 0.4487245 0.4077411 0.3654875 0.3235514 0.2837316
## [22] 0.2515254 0.2207263 0.1998491 0.1826889 0.1586367
##
## $freq
## [1] 0.003831418 0.007662835 0.011494253 0.015325670 0.019157088 0.022988506
## [7] 0.026819923 0.030651341 0.034482759 0.038314176 0.042145594 0.045977011
## [13] 0.049808429 0.053639847 0.057471264 0.061302682 0.065134100 0.068965517
## [19] 0.072796935 0.076628352 0.080459770 0.084291188 0.088122605 0.091954023
## [25] 0.095785441 0.099616858 0.103448276 0.107279693 0.111111111 0.114942529
## [31] 0.118773946 0.122605364 0.126436782 0.130268199 0.134099617 0.137931034
## [37] 0.141762452 0.145593870 0.149425287 0.153256705 0.157088123 0.160919540
## [43] 0.164750958 0.168582375 0.172413793 0.176245211 0.180076628 0.183908046
## [49] 0.187739464 0.191570881 0.195402299 0.199233716 0.203065134 0.206896552
## [55] 0.210727969 0.214559387 0.218390805 0.222222222 0.226053640 0.229885057
## [61] 0.233716475 0.237547893 0.241379310 0.245210728 0.249042146 0.252873563
## [67] 0.256704981 0.260536398 0.264367816 0.268199234 0.272030651 0.275862069
## [73] 0.279693487 0.283524904 0.287356322 0.291187739 0.295019157 0.298850575
## [79] 0.302681992 0.306513410 0.310344828 0.314176245 0.318007663 0.321839080
## [85] 0.325670498 0.329501916 0.333333333 0.337164751 0.340996169 0.344827586
## [91] 0.348659004 0.352490421 0.356321839 0.360153257 0.363984674 0.367816092
## [97] 0.371647510 0.375478927 0.379310345 0.383141762 0.386973180 0.390804598
## [103] 0.394636015 0.398467433 0.402298851 0.406130268 0.409961686 0.413793103
## [109] 0.417624521 0.421455939 0.425287356 0.429118774 0.432950192 0.436781609
## [115] 0.440613027 0.444444444 0.448275862 0.452107280 0.455938697 0.459770115
## [121] 0.463601533 0.467432950 0.471264368 0.475095785 0.478927203 0.482758621
## [127] 0.486590038 0.490421456 0.494252874 0.498084291
##
## $dbz
## [1] 12.34247354 12.11914326 11.74635018 11.22342293 10.54987428
## [6] 9.72603461 8.75418736 7.64056246 6.39872676 5.05498450
## [11] 3.65581433 2.27479289 1.01067362 -0.03604101 -0.80727694
## [16] -1.31903982 -1.64675193 -1.88122193 -2.09593187 -2.33839785
## [21] -2.63486737 -2.99735762 -3.42858191 -3.92430375 -4.47404643
## [26] -5.06141232 -5.66526486 -6.26269996 -6.83377528 -7.36642486
## [31] -7.85890296 -8.31787269 -8.75274349 -9.16904719 -9.56399873
## [36] -9.92632939 -10.24088770 -10.49642769 -10.69279641 -10.84330773
## [41] -10.97088671 -11.10048445 -11.25169391 -11.43398966 -11.64498097
## [46] -11.87120849 -12.09114324 -12.28018512 -12.41691364 -12.48879033
## [51] -12.49507403 -12.44586320 -12.35826980 -12.25194835 -12.14575162
## [56] -12.05602858 -11.99615702 -11.97665194 -12.00534780 -12.08739385
## [61] -12.22496696 -12.41668031 -12.65669968 -12.93364814 -13.22956668
## [66] -13.51955972 -13.77319751 -13.95882941 -14.05091009 -14.03806607
## [71] -13.92769631 -13.74414594 -13.52160997 -13.29588031 -13.09837508
## [76] -12.95336172 -12.87751581 -12.88058643 -12.96629340 -13.13302503
## [81] -13.37420317 -13.67834022 -14.02889961 -14.40414854 -14.77730940
## [86] -15.11749541 -15.39208435 -15.57106927 -15.63311292 -15.57151749
## [91] -15.39722851 -15.13692019 -14.82696395 -14.50622252 -14.21040074
## [96] -13.96897449 -13.80426228 -13.73169224 -13.76044971 -13.89400346
## [101] -14.13027252 -14.46138142 -14.87311844 -15.34443090 -15.84758554
## [106] -16.34986662 -16.81750858 -17.22153106 -17.54346560 -17.77806885
## [111] -17.93148745 -18.01620729 -18.04583417 -18.03203751 -17.98407988
## [116] -17.90993918 -17.81767355 -17.71608913 -17.61445144 -17.52152841
## [121] -17.44446679 -17.38791989 -17.35363279 -17.34050359 -17.34503321
## [126] -17.36203822 -17.38549708 -17.40940880 -17.42856219 -17.43914307
## $freq
## [1] 0.003831418 0.007662835 0.011494253 0.015325670 0.019157088 0.022988506
## [7] 0.026819923 0.030651341 0.034482759 0.038314176 0.042145594 0.045977011
## [13] 0.049808429 0.053639847 0.057471264 0.061302682 0.065134100 0.068965517
## [19] 0.072796935 0.076628352 0.080459770 0.084291188 0.088122605 0.091954023
## [25] 0.095785441 0.099616858 0.103448276 0.107279693 0.111111111 0.114942529
## [31] 0.118773946 0.122605364 0.126436782 0.130268199 0.134099617 0.137931034
## [37] 0.141762452 0.145593870 0.149425287 0.153256705 0.157088123 0.160919540
## [43] 0.164750958 0.168582375 0.172413793 0.176245211 0.180076628 0.183908046
## [49] 0.187739464 0.191570881 0.195402299 0.199233716 0.203065134 0.206896552
## [55] 0.210727969 0.214559387 0.218390805 0.222222222 0.226053640 0.229885057
## [61] 0.233716475 0.237547893 0.241379310 0.245210728 0.249042146 0.252873563
## [67] 0.256704981 0.260536398 0.264367816 0.268199234 0.272030651 0.275862069
## [73] 0.279693487 0.283524904 0.287356322 0.291187739 0.295019157 0.298850575
## [79] 0.302681992 0.306513410 0.310344828 0.314176245 0.318007663 0.321839080
## [85] 0.325670498 0.329501916 0.333333333 0.337164751 0.340996169 0.344827586
## [91] 0.348659004 0.352490421 0.356321839 0.360153257 0.363984674 0.367816092
## [97] 0.371647510 0.375478927 0.379310345 0.383141762 0.386973180 0.390804598
## [103] 0.394636015 0.398467433 0.402298851 0.406130268 0.409961686 0.413793103
## [109] 0.417624521 0.421455939 0.425287356 0.429118774 0.432950192 0.436781609
## [115] 0.440613027 0.444444444 0.448275862 0.452107280 0.455938697 0.459770115
## [121] 0.463601533 0.467432950 0.471264368 0.475095785 0.478927203 0.482758621
## [127] 0.486590038 0.490421456 0.494252874 0.498084291
##
## $pzgram
## [1] 12.34247354 12.11914326 11.74635018 11.22342293 10.54987428
## [6] 9.72603461 8.75418736 7.64056246 6.39872676 5.05498450
## [11] 3.65581433 2.27479289 1.01067362 -0.03604101 -0.80727694
## [16] -1.31903982 -1.64675193 -1.88122193 -2.09593187 -2.33839785
## [21] -2.63486737 -2.99735762 -3.42858191 -3.92430375 -4.47404643
## [26] -5.06141232 -5.66526486 -6.26269996 -6.83377528 -7.36642486
## [31] -7.85890296 -8.31787269 -8.75274349 -9.16904719 -9.56399873
## [36] -9.92632939 -10.24088770 -10.49642769 -10.69279641 -10.84330773
## [41] -10.97088671 -11.10048445 -11.25169391 -11.43398966 -11.64498097
## [46] -11.87120849 -12.09114324 -12.28018512 -12.41691364 -12.48879033
## [51] -12.49507403 -12.44586320 -12.35826980 -12.25194835 -12.14575162
## [56] -12.05602858 -11.99615702 -11.97665194 -12.00534780 -12.08739385
## [61] -12.22496696 -12.41668031 -12.65669968 -12.93364814 -13.22956668
## [66] -13.51955972 -13.77319751 -13.95882941 -14.05091009 -14.03806607
## [71] -13.92769631 -13.74414594 -13.52160997 -13.29588031 -13.09837508
## [76] -12.95336172 -12.87751581 -12.88058643 -12.96629340 -13.13302503
## [81] -13.37420317 -13.67834022 -14.02889961 -14.40414854 -14.77730940
## [86] -15.11749541 -15.39208435 -15.57106927 -15.63311292 -15.57151749
## [91] -15.39722851 -15.13692019 -14.82696395 -14.50622252 -14.21040074
## [96] -13.96897449 -13.80426228 -13.73169224 -13.76044971 -13.89400346
## [101] -14.13027252 -14.46138142 -14.87311844 -15.34443090 -15.84758554
## [106] -16.34986662 -16.81750858 -17.22153106 -17.54346560 -17.77806885
## [111] -17.93148745 -18.01620729 -18.04583417 -18.03203751 -17.98407988
## [116] -17.90993918 -17.81767355 -17.71608913 -17.61445144 -17.52152841
## [121] -17.44446679 -17.38791989 -17.35363279 -17.34050359 -17.34503321
## [126] -17.36203822 -17.38549708 -17.40940880 -17.42856219 -17.43914307
The ACF plots shows a slowly dampening autocorrelation, along with the
spectral density, there may be some frequency with the five peaks in the
spectral density, however there is strong evidence of wandering behavior
and stationary.
Now that we have a sense of the data lets see what a base time series model may yield
#First put the object into a ts object
ts_nly_open = ts(df.adj$Open, frequency = 1)
ts_nly_open.eco = ts(finaldata$Open, frequency = 1)Fit at least one model from each of the following four categories (provide all plots and tables needed to ID these models: acfs, spectral density, factor tables, etc.):
a. ARMA / ARIMA / ARUMA / Signal Plus Noise (univariate analysis)
aic5.wge(ts_nly_open, p = 0:10, q = 0:10) #picks a 6:6 model, lets difference the data first## ---------WORKING... PLEASE WAIT...
##
##
## Error in aic calculation at 7 7
## Five Smallest Values of aic
## p q aic
## 6 6 -3.078345
## 8 7 -3.050791
## 8 2 -3.048955
## 5 2 -3.045016
## 9 2 -3.041199aic5.wge(ts_nly_open.eco, p = 0:10, q = 0:10) #picks a 6:5 model, lets difference the data first## ---------WORKING... PLEASE WAIT...
##
##
## Error in aic calculation at 8 9
## Five Smallest Values of aic
## p q aic
## 6 5 -3.086153
## 6 6 -3.070210
## 8 2 -3.044905
## 6 4 -3.041284
## 5 2 -3.041058#Original Dataset
diff_ts = artrans.wge(ts_nly_open, 1) # we get an acf plot at lag 1 as opposed to lag 0, lets difference the differences data
diff_two_ts = artrans.wge(diff_ts,1) # the acf lag has another pronounced at lag 2, however the first differences data may be better suited. We'll proceed with the first differenced data set.
#New Data Set
diff_ts.eco = artrans.wge(ts_nly_open.eco, 1) # we get an acf plot at lag 1 as opposed to lag 0, lets difference the differences data
diff_two_ts.eco = artrans.wge(diff_ts.eco,1) # the acf lag has another pronounced at lag 2, however the first differences data may be better suited. We'll proceed with the first differenced data set.
#Original Data Set
aic5.wge(diff_ts, p = 0:10, q = 0:10, type = "aic") # at iteration 80, AIC picks AR(7)MA(2) model## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aic
## p q aic
## 7 2 -3.036156
## 4 2 -3.031027
## 7 3 -3.017738
## 2 9 -3.015115
## 3 0 -3.014866aic5.wge(diff_ts, p= 0:10, q=0:10, type ="bic") # at iteration 75, BIC picks AR(0)MA(0) model## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of bic
## p q bic
## 0 0 -2.996849
## 0 1 -2.976414
## 1 0 -2.976223
## 2 0 -2.967971
## 0 2 -2.967646#New Data Set
aic5.wge(diff_ts.eco, p = 0:10, q = 0:10, type = "aic") # at iteration 80, AIC picks AR(7)MA(2) model## ---------WORKING... PLEASE WAIT...
##
##
## Error in aic calculation at 6 6
## Error in aic calculation at 6 10
## Error in aic calculation at 7 8
## Five Smallest Values of aic
## p q aic
## 7 2 -3.032431
## 4 2 -3.027806
## 7 3 -3.013965
## 3 3 -3.011473
## 3 0 -3.010912aic5.wge(diff_ts.eco, p= 0:10, q=0:10, type ="bic") # at iteration 75, BIC picks AR(0)MA(0) model## ---------WORKING... PLEASE WAIT...
##
##
## Error in aic calculation at 6 6
## Error in aic calculation at 6 10
## Error in aic calculation at 7 8
## Five Smallest Values of bic
## p q bic
## 0 0 -2.993063
## 0 1 -2.972567
## 1 0 -2.972376
## 2 0 -2.963938
## 0 2 -2.963613Noting that the BIC choose a AR0MA0 model, however in the updated it suggests an MA(1) model for its second choice, this tells us that we have likely high degrees of wandering even after differences the data, we’ll model both, but give the BIC a straight seasonal component
#Original Data Set
#AIC Model AR(7)MA(2) model
aic_est = est.arma.wge(diff_ts, p = 7, q = 2)##
##
## Coefficients of AR polynomial:
## -1.4236 -1.0081 -0.0554 0.0271 0.0002 -0.1162 -0.1176
##
## AR Factor Table
## Factor Roots Abs Recip System Freq
## 1+1.4581B+0.9751B^2 -0.7476+-0.6830i 0.9875 0.3822
## 1+0.3322B+0.4935B^2 -0.3365+-1.3831i 0.7025 0.2880
## 1+0.6515B -1.5349 0.6515 0.5000
## 1-1.0182B+0.3750B^2 1.3575+-0.9075i 0.6124 0.0938
##
##
##
##
## Coefficients of MA polynomial:
## -1.4456 -0.9218
##
## MA FACTOR TABLE
## Factor Roots Abs Recip System Freq
## 1+1.4456B+0.9218B^2 -0.7841+-0.6856i 0.9601 0.3857
##
## #BIC Model AR(0)MA(0) model
bic_est = est.arma.wge(diff_ts, p = 0, q = 0)
#New Data Set
#AIC Model AR(7)MA(2) model
aic_est.eco = est.arma.wge(diff_ts.eco, p = 7, q = 2)##
##
## Coefficients of AR polynomial:
## -1.4233 -1.0093 -0.0568 0.0270 0.0011 -0.1166 -0.1186
##
## AR Factor Table
## Factor Roots Abs Recip System Freq
## 1+1.4584B+0.9756B^2 -0.7474+-0.6829i 0.9877 0.3822
## 1+0.3316B+0.4958B^2 -0.3344+-1.3802i 0.7042 0.2878
## 1+0.6524B -1.5328 0.6524 0.5000
## 1-1.0192B+0.3757B^2 1.3566+-0.9065i 0.6129 0.0938
##
##
##
##
## Coefficients of MA polynomial:
## -1.4460 -0.9222
##
## MA FACTOR TABLE
## Factor Roots Abs Recip System Freq
## 1+1.4460B+0.9222B^2 -0.7840+-0.6853i 0.9603 0.3857
##
## #BIC Model AR(0)MA(0) model
bic_est.eco = est.arma.wge(diff_ts.eco, p = 0, q = 0)b. VAR with at least one explanatory variable
# Create a data frame with all four predictors (Open, Close, High, Low columns)
tnlyx <- data.frame(Open = ts(df.adj$Open), Close = ts(df.adj$Close), High = ts(df.adj$High), Low = ts(df.adj$Low), frequency = 1)
# Remove rows with NA values from the entire data frame
df_clean <- tnlyx[complete.cases(tnlyx), ]
# Apply VARSelect to select the optimal lag order
var_order <- VARselect(df_clean, lag.max = 10, type = "both")
# Get the optimal lag order selected by AIC, HQIC, and SC
optimal_lag <- var_order$selection
optimal_lag## AIC(n) HQ(n) SC(n) FPE(n)
## 1 1 1 1# Create a data frame with all four variables (Open, Close, High, Low columns)
data <- data.frame(Open = df.adj$Open, Close = df.adj$Close, High = df.adj$High, Low = df.adj$Low)
# Set the lag order you want to use
lag_order <- 1
# Lag the other three variables (Close, High, Low) in the data frame
lagged_data <- data %>% mutate_at(vars(-Open), lag, n = lag_order)
# Remove rows with NA values from the lagged_data
lagged_data <- na.omit(lagged_data)
# Fit the VAR model using the lagged_data
var_model <- VAR(lagged_data, p = lag_order, type = "both")
var_model##
## VAR Estimation Results:
## =======================
##
## Estimated coefficients for equation Open:
## =========================================
## Call:
## Open = Open.l1 + Close.l1 + High.l1 + Low.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 const
## 0.9267299491 -0.0725082332 0.0518762053 0.0563128408 0.4193964134
## trend
## -0.0001087287
##
##
## Estimated coefficients for equation Close:
## ==========================================
## Call:
## Close = Open.l1 + Close.l1 + High.l1 + Low.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 const
## 0.8957877004 -0.0129277781 0.0588175712 0.0183600471 0.4523733878
## trend
## -0.0002184793
##
##
## Estimated coefficients for equation High:
## =========================================
## Call:
## High = Open.l1 + Close.l1 + High.l1 + Low.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 const trend
## 0.940172100 -0.032746836 0.147284505 -0.074003928 0.356700949 -0.000180586
##
##
## Estimated coefficients for equation Low:
## ========================================
## Call:
## Low = Open.l1 + Close.l1 + High.l1 + Low.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 const
## 1.0683737037 -0.1966631921 -0.0835850892 0.1845078280 0.1740256553
## trend
## -0.0000982939summary(var_model)##
## VAR Estimation Results:
## =========================
## Endogenous variables: Open, Close, High, Low
## Deterministic variables: both
## Sample size: 260
## Log Likelihood: 970.857
## Roots of the characteristic polynomial:
## 0.9642 0.2653 0.05207 0.03593
## Call:
## VAR(y = lagged_data, p = lag_order, type = "both")
##
##
## Estimation results for equation Open:
## =====================================
## Open = Open.l1 + Close.l1 + High.l1 + Low.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 0.9267299 0.2720113 3.407 0.000764 ***
## Close.l1 -0.0725082 0.3009176 -0.241 0.809783
## High.l1 0.0518762 0.1076667 0.482 0.630347
## Low.l1 0.0563128 0.1215762 0.463 0.643625
## const 0.4193964 0.2000355 2.097 0.037017 *
## trend -0.0001087 0.0001869 -0.582 0.561207
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.2214 on 254 degrees of freedom
## Multiple R-Squared: 0.9195, Adjusted R-squared: 0.9179
## F-statistic: 579.9 on 5 and 254 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Close:
## ======================================
## Close = Open.l1 + Close.l1 + High.l1 + Low.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 0.8957877 0.2612343 3.429 0.000707 ***
## Close.l1 -0.0129278 0.2889954 -0.045 0.964355
## High.l1 0.0588176 0.1034010 0.569 0.569974
## Low.l1 0.0183600 0.1167594 0.157 0.875175
## const 0.4523734 0.1921102 2.355 0.019296 *
## trend -0.0002185 0.0001795 -1.217 0.224609
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.2127 on 254 degrees of freedom
## Multiple R-Squared: 0.9249, Adjusted R-squared: 0.9234
## F-statistic: 625.2 on 5 and 254 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation High:
## =====================================
## High = Open.l1 + Close.l1 + High.l1 + Low.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 0.9401721 0.1534984 6.125 3.43e-09 ***
## Close.l1 -0.0327468 0.1698105 -0.193 0.84724
## High.l1 0.1472845 0.0607573 2.424 0.01604 *
## Low.l1 -0.0740039 0.0686065 -1.079 0.28176
## const 0.3567009 0.1128818 3.160 0.00177 **
## trend -0.0001806 0.0001055 -1.712 0.08804 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.125 on 254 degrees of freedom
## Multiple R-Squared: 0.9738, Adjusted R-squared: 0.9733
## F-statistic: 1889 on 5 and 254 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Low:
## ====================================
## Low = Open.l1 + Close.l1 + High.l1 + Low.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 1.068e+00 2.020e-01 5.289 2.65e-07 ***
## Close.l1 -1.967e-01 2.234e-01 -0.880 0.380
## High.l1 -8.359e-02 7.995e-02 -1.045 0.297
## Low.l1 1.845e-01 9.028e-02 2.044 0.042 *
## const 1.740e-01 1.485e-01 1.172 0.242
## trend -9.829e-05 1.388e-04 -0.708 0.479
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.1644 on 254 degrees of freedom
## Multiple R-Squared: 0.9551, Adjusted R-squared: 0.9542
## F-statistic: 1080 on 5 and 254 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## Open Close High Low
## Open 0.04903 0.04583 0.02054 0.02844
## Close 0.04583 0.04522 0.01999 0.02817
## High 0.02054 0.01999 0.01561 0.01157
## Low 0.02844 0.02817 0.01157 0.02703
##
## Correlation matrix of residuals:
## Open Close High Low
## Open 1.0000 0.9733 0.7423 0.7811
## Close 0.9733 1.0000 0.7525 0.8058
## High 0.7423 0.7525 1.0000 0.5631
## Low 0.7811 0.8058 0.5631 1.0000#xtract the estimated coefficients from the var_model list
coefficients <- coef(var_model)[['Open']] # Replace 'Open' with the desired variable
# Extract the estimated coefficients and convert to numeric format
coefficients <- as.numeric(coefficients)
# Extract the standard errors of the coefficients
standard_errors <- sqrt(diag(vcov(var_model)))
# Calculate the confidence intervals (95% by default)
lower_ci <- coefficients - 1.96 * standard_errors
upper_ci <- coefficients + 1.96 * standard_errors
# Create a data frame to store the results
confidence_intervals <- data.frame(
Coefficients = coefficients,
Lower_CI = lower_ci,
Upper_CI = upper_ci
)
print(confidence_intervals)## Coefficients Lower_CI Upper_CI
## Open:(Intercept) 0.9267299491 0.5346604061 1.3187994922
## Open:Open.l1 -0.0725082332 -0.6056503098 0.4606338434
## Open:Close.l1 0.0518762053 -0.5379222888 0.6416746994
## Open:High.l1 0.0563128408 -0.1547139621 0.2673396437
## Open:Low.l1 0.4193964134 0.1811070801 0.6576857468
## Open:trend -0.0001087287 -0.0004750103 0.0002575530
## Close:(Intercept) 0.2720112636 -0.1045246403 0.6485471674
## Close:Open.l1 0.3009175990 -0.2111016007 0.8129367988
## Close:Close.l1 0.1076667362 -0.4587641764 0.6740976488
## Close:High.l1 0.1215761905 -0.0810898145 0.3242421955
## Close:Low.l1 0.2000354811 -0.0288129237 0.4288838859
## Close:trend 0.0001868784 -0.0001648913 0.0005386481
## High:(Intercept) 3.4069543185 3.1857059705 3.6282026666
## High:Open.l1 -0.2409571039 -0.5418139438 0.0598997360
## High:Close.l1 0.4818220291 0.1489934663 0.8146505919
## High:High.l1 0.4631897130 0.3441054016 0.5822740244
## High:Low.l1 2.0966101166 1.9621413156 2.2310789175
## High:trend -0.5818151923 -0.5820218883 -0.5816084963
## Low:(Intercept) 0.0007635957 -0.2903701649 0.2918973564
## Low:Open.l1 0.8097827491 0.4138946714 1.2056708268
## Low:Close.l1 0.6303472613 0.1923885964 1.0683059262
## Low:High.l1 0.6436250086 0.4869256989 0.8003243182
## Low:Low.l1 0.0370173831 -0.1399258939 0.2139606600
## Low:trend 0.5612071811 0.5609351963 0.5614791659Observations
Estimated coefficients for equation Open:
- Open.l1: Coefficient for the lagged ‘Open’ variable (Open[t-1]).
- Close.l1: Coefficient for the lagged ‘Close’ variable (Close[t-1]).
- High.l1: Coefficient for the lagged ‘High’ variable (High[t-1]).
- Low.l1: Coefficient for the lagged ‘Low’ variable (Low[t-1]).
- const: Coefficient for the constant term (intercept).
- trend: Coefficient for the trend term.
** A one-unit increase in the lagged ‘Open’ variable (Open[t-1]) leads to a 0.9267 increase in the current ‘Open’ variable (Open[t]).
** A one-unit increase in the lagged ‘Close’ variable (Close[t-1]) leads to a -0.0725 decrease in the current ‘Open’ variable (Open[t]).
**A one-unit increase in the lagged ‘High’ variable (High[t-1]) leads to a 0.0519 increase in the current ‘Open’ variable (Open[t]).
**A one-unit increase in the lagged ‘Low’ variable (Low[t-1]) leads to a 0.0563 increase in the current ‘Open’ variable (Open[t]).
The constant term (intercept) and trend do not have lagged values as they are constant across time.
The equation’s multiple R-squared and adjusted R-squared are 0.9195 and 0.9179, respectively, indicating a good fit.
The F-statistic and its p-value suggest that the overall equation is significant.
# Fit the VAR model using the lagged_data
VAR_SM2 = VAR(lagged_data,lag.max = 5, type = "both")
pred.short = predict(VAR_SM2,n.ahead = 12)
pred.short$fcst$Open[,1]## [1] 9.970557 9.984902 9.998284 10.010942 10.023007 10.034528 10.045531
## [8] 10.056037 10.066064 10.075630 10.084750 10.093442plot(data$Open, type = "l")
lines(seq(251,262,1),pred.short$fcst$Open[,1],col = "red")
var_ase_short_horizon = mean((data$Open[251:262] - pred.short$fcst$Open[,1])^2)
var_ase_short_horizon## [1] 0.06697107# Fit the VAR model using the lagged_data
VAR_SM2 = VAR(lagged_data,lag.max = 15, type = "both")
pred.long = predict(VAR_SM2,n.ahead = 20)
pred.long$fcst$Open[,1]## [1] 9.970557 9.984902 9.998284 10.010942 10.023007 10.034528 10.045531
## [8] 10.056037 10.066064 10.075630 10.084750 10.093442 10.101720 10.109599
## [15] 10.117093 10.124217 10.130983 10.137405 10.143494 10.149263plot(data$Open, type = "l")
lines(seq(243,262,1),pred.long$fcst$Open[,1],col = "red")
var_ase_long_horizon = mean((data$Open[243:262] - pred.long$fcst$Open[,1])^2)
var_ase_long_horizon ## [1] 0.1551258b PT II Make supplemental model with addtional economic features
# Create a data frame with all four predictors (Open, Close, High, Low columns)
tnlyx.eco <- data.frame(Open = ts(finaldata$Open), Close = ts(finaldata$Close), High = ts(finaldata$High), Low = ts(finaldata$Low), CPI = ts(finaldata$CPI_Score), Morg = ts(finaldata$MORTGAGE30US), frequency = 1)
# Remove rows with NA values from the entire data frame
df_clean.eco <- tnlyx.eco[complete.cases(tnlyx.eco), ]
# Apply VARSelect to select the optimal lag order
var_order.eco <- VARselect(df_clean.eco, lag.max = 10, type = "both")
# Get the optimal lag order selected by AIC, HQIC, and SC
optimal_lag.eco <- var_order.eco$selection
optimal_lag.eco## AIC(n) HQ(n) SC(n) FPE(n)
## 3 1 1 3It’s interesting to not that the ideal model is still looking at a lag 1 model from both the HQIC & SC criterion, however with the additional economic features added both the AIC & FPE are suggesting an optimal lag order of 3.
# Create a data frame with all six variables (Open, Close, High, Low, CPI Score, Monthly Mortgage Rate columns)
data.eco <- data.frame(Open = finaldata$Open, Close = finaldata$Close, High = finaldata$High, Low = finaldata$Low, Cpi = finaldata$CPI_Score, Morg = finaldata$MORTGAGE30US)
# Set the lag order you want to use, we will still use optimal lag order of 1 rather than suggested at 3
lag_order <- 1
# Lag the other three variables (Close, High, Low) in the data frame
lagged_data.eco <- data.eco %>% mutate_at(vars(-Open), lag, n = lag_order)
# Remove rows with NA values from the lagged_data
lagged_data.eco <- na.omit(lagged_data.eco)
# Fit the VAR model using the lagged_data
var_model.eco <- VAR(lagged_data.eco, p = lag_order, type = "both")
var_model.eco##
## VAR Estimation Results:
## =======================
##
## Estimated coefficients for equation Open:
## =========================================
## Call:
## Open = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 Cpi.l1
## 0.9199181459 -0.0697147839 0.0494788754 0.0684517737 -0.0048785930
## Morg.l1 const trend
## 0.0541031352 1.2845817216 0.0001719648
##
##
## Estimated coefficients for equation Close:
## ==========================================
## Call:
## Close = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 Cpi.l1 Morg.l1
## 0.892297516 -0.014609455 0.055702867 0.033061355 -0.005939828 0.051638085
## const trend
## 1.567338587 0.000136305
##
##
## Estimated coefficients for equation High:
## =========================================
## Call:
## High = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 Cpi.l1
## 0.9419165019 -0.0455695840 0.1404127191 -0.0440088704 -0.0121763466
## Morg.l1 const trend
## 0.0789380537 2.7593275606 0.0005723216
##
##
## Estimated coefficients for equation Low:
## ========================================
## Call:
## Low = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 Cpi.l1
## 1.0552822336 -0.1765185131 -0.0781046211 0.1666980653 0.0073008711
## Morg.l1 const trend
## -0.0078037101 -1.4414927715 -0.0005927528
##
##
## Estimated coefficients for equation Cpi:
## ========================================
## Call:
## Cpi = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 Cpi.l1 Morg.l1
## -0.026034978 0.089520361 0.078944546 -0.087146208 0.950286092 0.241467622
## const trend
## 10.054341812 0.003314133
##
##
## Estimated coefficients for equation Morg:
## =========================================
## Call:
## Morg = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Open.l1 Close.l1 High.l1 Low.l1 Cpi.l1
## 0.0519922684 -0.1566944448 0.0432570834 0.0494048119 0.0044952797
## Morg.l1 const trend
## 0.9771518634 -0.8346091457 -0.0001934743summary(var_model.eco)##
## VAR Estimation Results:
## =========================
## Endogenous variables: Open, Close, High, Low, Cpi, Morg
## Deterministic variables: both
## Sample size: 259
## Log Likelihood: 1328.41
## Roots of the characteristic polynomial:
## 0.9968 0.9475 0.9475 0.2369 0.05038 0.03913
## Call:
## VAR(y = lagged_data.eco, p = lag_order, type = "both")
##
##
## Estimation results for equation Open:
## =====================================
## Open = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 0.9199181 0.2737873 3.360 0.000901 ***
## Close.l1 -0.0697148 0.3034541 -0.230 0.818483
## High.l1 0.0494789 0.1083410 0.457 0.648285
## Low.l1 0.0684518 0.1253812 0.546 0.585586
## Cpi.l1 -0.0048786 0.0116480 -0.419 0.675696
## Morg.l1 0.0541031 0.0653211 0.828 0.408308
## const 1.2845817 2.4234768 0.530 0.596541
## trend 0.0001720 0.0007634 0.225 0.821949
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.2224 on 251 degrees of freedom
## Multiple R-Squared: 0.9194, Adjusted R-squared: 0.9171
## F-statistic: 409 on 7 and 251 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Close:
## ======================================
## Close = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 0.8922975 0.2629871 3.393 0.000803 ***
## Close.l1 -0.0146095 0.2914835 -0.050 0.960066
## High.l1 0.0557029 0.1040672 0.535 0.592945
## Low.l1 0.0330614 0.1204352 0.275 0.783914
## Cpi.l1 -0.0059398 0.0111885 -0.531 0.595968
## Morg.l1 0.0516381 0.0627444 0.823 0.411294
## const 1.5673386 2.3278763 0.673 0.501382
## trend 0.0001363 0.0007332 0.186 0.852679
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.2136 on 251 degrees of freedom
## Multiple R-Squared: 0.9247, Adjusted R-squared: 0.9227
## F-statistic: 440.6 on 7 and 251 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation High:
## =====================================
## High = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 0.9419165 0.1531482 6.150 3.03e-09 ***
## Close.l1 -0.0455696 0.1697429 -0.268 0.7886
## High.l1 0.1404127 0.0606026 2.317 0.0213 *
## Low.l1 -0.0440089 0.0701344 -0.627 0.5309
## Cpi.l1 -0.0121763 0.0065156 -1.869 0.0628 .
## Morg.l1 0.0789381 0.0365386 2.160 0.0317 *
## const 2.7593276 1.3556183 2.035 0.0429 *
## trend 0.0005723 0.0004270 1.340 0.1813
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.1244 on 251 degrees of freedom
## Multiple R-Squared: 0.9742, Adjusted R-squared: 0.9735
## F-statistic: 1355 on 7 and 251 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Low:
## ====================================
## Low = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 1.0552822 0.2030237 5.198 4.18e-07 ***
## Close.l1 -0.1765185 0.2250227 -0.784 0.4335
## High.l1 -0.0781046 0.0803389 -0.972 0.3319
## Low.l1 0.1666981 0.0929749 1.793 0.0742 .
## Cpi.l1 0.0073009 0.0086375 0.845 0.3988
## Morg.l1 -0.0078037 0.0484381 -0.161 0.8721
## const -1.4414928 1.7971001 -0.802 0.4232
## trend -0.0005928 0.0005661 -1.047 0.2960
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.1649 on 251 degrees of freedom
## Multiple R-Squared: 0.9552, Adjusted R-squared: 0.954
## F-statistic: 765.1 on 7 and 251 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Cpi:
## ====================================
## Cpi = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 -0.026035 0.439065 -0.059 0.95276
## Close.l1 0.089520 0.486641 0.184 0.85420
## High.l1 0.078945 0.173743 0.454 0.64995
## Low.l1 -0.087146 0.201070 -0.433 0.66509
## Cpi.l1 0.950286 0.018680 50.873 < 2e-16 ***
## Morg.l1 0.241468 0.104754 2.305 0.02198 *
## const 10.054342 3.886460 2.587 0.01024 *
## trend 0.003314 0.001224 2.707 0.00725 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.3566 on 251 degrees of freedom
## Multiple R-Squared: 0.9959, Adjusted R-squared: 0.9958
## F-statistic: 8752 on 7 and 251 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Morg:
## =====================================
## Morg = Open.l1 + Close.l1 + High.l1 + Low.l1 + Cpi.l1 + Morg.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## Open.l1 0.0519923 0.0538543 0.965 0.33526
## Close.l1 -0.1566944 0.0596898 -2.625 0.00919 **
## High.l1 0.0432571 0.0213108 2.030 0.04343 *
## Low.l1 0.0494048 0.0246626 2.003 0.04623 *
## Cpi.l1 0.0044953 0.0022912 1.962 0.05087 .
## Morg.l1 0.9771519 0.0128488 76.050 < 2e-16 ***
## const -0.8346091 0.4767009 -1.751 0.08120 .
## trend -0.0001935 0.0001502 -1.289 0.19875
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.04374 on 251 degrees of freedom
## Multiple R-Squared: 0.9839, Adjusted R-squared: 0.9834
## F-statistic: 2186 on 7 and 251 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## Open Close High Low Cpi Morg
## Open 0.049450 0.046231 0.0205802 0.0287639 -0.001456 -0.0011934
## Close 0.046231 0.045625 0.0200324 0.0285248 -0.001350 -0.0011136
## High 0.020580 0.020032 0.0154725 0.0117992 -0.003395 -0.0006113
## Low 0.028764 0.028525 0.0117992 0.0271913 -0.001470 -0.0005077
## Cpi -0.001456 -0.001350 -0.0033946 -0.0014701 0.127173 -0.0018364
## Morg -0.001193 -0.001114 -0.0006113 -0.0005077 -0.001836 0.0019133
##
## Correlation matrix of residuals:
## Open Close High Low Cpi Morg
## Open 1.00000 0.97331 0.74403 0.78442 -0.01836 -0.12269
## Close 0.97331 1.00000 0.75396 0.80985 -0.01773 -0.11919
## High 0.74403 0.75396 1.00000 0.57525 -0.07653 -0.11235
## Low 0.78442 0.80985 0.57525 1.00000 -0.02500 -0.07038
## Cpi -0.01836 -0.01773 -0.07653 -0.02500 1.00000 -0.11773
## Morg -0.12269 -0.11919 -0.11235 -0.07038 -0.11773 1.00000The R-squared and Adjusted R-squared values provide a measure of how well the model fits the data. It indicates the proportion of variance in the target variable that is explained by the independent variables. For example, in the equation for “Open,” the R-squared value is approximately 0.9194, indicating that about 91.94% of the variation in “Open” can be explained by the lagged values of all variables.
The F-statistic tests the overall significance of the model. It assesses whether at least one of the independent variables has a statistically significant impact on the dependent variable. For instance, in the equation for “Open,” the F-statistic is 409, and the p-value is virtually zero (p-value: < 2.2e-16). This indicates that the overall model is highly significant.
Overall, the VAR model results show that several lagged variables have significant effects on each target variable (Open, Close, High, Low, Cpi, Morg), indicating dynamic relationships among the variables in the system. Make sure to validate the model assumptions, such as the independence and normality of residuals, to ensure the model’s reliability. It is important to note that only lag 1 of “Open” is the only statistically significant feature contained in the model.
# Extract the estimated coefficients from the var_model list
coefficients.eco <- coef(var_model.eco)[['Open']] # Replace 'Open' with the desired variable
coefficients.eco## Estimate Std. Error t value Pr(>|t|)
## Open.l1 0.9199181459 0.2737873451 3.3599732 0.000900853
## Close.l1 -0.0697147839 0.3034540750 -0.2297375 0.818482851
## High.l1 0.0494788754 0.1083409576 0.4566959 0.648284573
## Low.l1 0.0684517737 0.1253811764 0.5459494 0.585585641
## Cpi.l1 -0.0048785930 0.0116480371 -0.4188339 0.675695694
## Morg.l1 0.0541031352 0.0653211450 0.8282637 0.408308015
## const 1.2845817216 2.4234767986 0.5300574 0.596540734
## trend 0.0001719648 0.0007633562 0.2252746 0.821948969# Extract the p-values of the coefficients
p_values.eco <- summary(var_model.eco)$Open$p.value # Replace 'Open' with the desired variable
# Check the significance of coefficients (optional)
print(p_values.eco)## NULL# Extract the estimated coefficients and convert to numeric format
coefficients.eco <- as.numeric(coefficients.eco)
# Extract the standard errors of the coefficients
standard_errors.eco <- sqrt(diag(vcov(var_model.eco)))
# Filter out non-significant coefficients
significant_indices <- p_values.eco <= 0.05
coefficients.eco <- coefficients.eco[significant_indices]
standard_errors.eco <- standard_errors.eco[significant_indices]
# Calculate the confidence intervals (95% by default) for significant coefficients
lower_ci.eco <- coefficients.eco - 1.96 * standard_errors.eco
upper_ci.eco <- coefficients.eco + 1.96 * standard_errors.eco
# Create a data frame to store the results
confidence_intervals.eco <- data.frame(
Coefficients.eco = coefficients.eco,
Lower_CI.eco = lower_ci.eco,
Upper_CI.eco = upper_ci.eco
)
print(confidence_intervals.eco)## [1] Coefficients.eco Lower_CI.eco Upper_CI.eco
## <0 rows> (or 0-length row.names)print(p_values.eco)## NULL# Fit the VAR model using the lagged_data
VAR_SM2.eco = VAR(lagged_data.eco,lag.max = 5, type = "both")
pred.short.eco = predict(VAR_SM2.eco,n.ahead = 12)
pred.short.eco$fcst$Open[,1]## [1] 9.997905 10.051699 10.111398 10.167951 10.222868 10.275086 10.323564
## [8] 10.368556 10.410470 10.449502 10.485816 10.519600plot(data.eco$Open, type = "l")
lines(seq(250,261,1),pred.short$fcst$Open[,1],col = "red")
var_ase_short_horizon.eco = mean((data.eco$Open[250:261] - pred.short.eco$fcst$Open[,1])^2)
var_ase_short_horizon.eco## [1] 0.2019013# Fit the VAR model using the lagged_data
VAR_SM2.eco = VAR(lagged_data.eco,lag.max = 15, type = "both")
pred.long.eco = predict(VAR_SM2.eco,n.ahead = 20)
pred.long.eco$fcst$Open[,1]## [1] 9.997905 10.051699 10.111398 10.167951 10.222868 10.275086 10.323564
## [8] 10.368556 10.410470 10.449502 10.485816 10.519600 10.551043 10.580318
## [15] 10.607587 10.633001 10.656701 10.678819 10.699474 10.718779plot(data.eco$Open, type = "l")
lines(seq(242,261,1),pred.long.eco$fcst$Open[,1],col = "red")
var_ase_long_horizon.eco = mean((data.eco$Open[242:261] - pred.long$fcst$Open[,1])^2)
var_ase_long_horizon.eco ## [1] 0.16422c. Neural Network (mlp) - short horizon
#MLP
NLYsmall = df.adj[1:250,]
NLYsmallDF = data.frame(Close = ts(NLYsmall$Close), High = ts(NLYsmall$High), Low = ts(NLYsmall$Low))
#Using forecast Open
fit.mlp.Close = mlp(ts(NLYsmallDF$Close),reps = 50, comb = "mean")
fit.mlp.High = mlp(ts(NLYsmallDF$High),reps = 50, comb = "mean")
fit.mlp.Low = mlp(ts(NLYsmallDF$Low),reps = 50, comb = "mean")
#Forecast the explainble features
fore.mlp.Close = forecast(fit.mlp.Close, h = 12)
fore.mlp.High = forecast(fit.mlp.High, h = 12)
fore.mlp.Low = forecast(fit.mlp.Low, h = 12)
#plot(fore.mlp.Open) # plot the forecasts
NLYsmallDF_fore = data.frame(Close = ts(fore.mlp.Close$mean), High = ts(fore.mlp.High$mean), Low = ts(fore.mlp.Low$mean))
NLYsmallDF_fore## Close High Low
## t+1 10.50384 10.77509 10.43800
## t+2 10.52037 10.75983 10.43602
## t+3 10.50798 10.75668 10.43408
## t+4 10.50913 10.75586 10.43217
## t+5 10.51536 10.75123 10.43029
## t+6 10.51737 10.74645 10.42844
## t+7 10.51895 10.74213 10.42662
## t+8 10.52174 10.73772 10.42484
## t+9 10.52485 10.73323 10.42309
## t+10 10.52771 10.72864 10.42136
## t+11 10.53048 10.72390 10.41967
## t+12 10.53348 10.71908 10.41800fit.mlp = mlp(ts(NLYsmall$Open),reps = 50,comb = "mean",hd.auto.type = "cv",xreg = NLYsmallDF) #sensitive to initial values, first 50 iterations
fit.mlp## MLP fit with 7 hidden nodes and 50 repetitions.
## Univariate lags: (2,3)
## 3 regressors included.
## - Regressor 1 lags: (1,2)
## - Regressor 2 lags: (3)
## - Regressor 3 lags: (2)
## Forecast combined using the mean operator.
## MSE: 0.0026.plot(fit.mlp)
NLYDF = data.frame(Close = ts(c(NLYsmallDF$Close,NLYsmallDF_fore$Close)), High = ts(c(NLYsmallDF$High, NLYsmallDF_fore$High)),Low = ts(c(NLYsmallDF$Low, NLYsmallDF_fore$Low)))
fore.mlp.short = forecast(fit.mlp, h = 12, xreg = NLYDF)
plot(fore.mlp.short)
plot(df.adj$Open, type = "l")
lines(seq(251,262,1),fore.mlp.short$mean,col = "blue")
MLP_SH_ASE = mean((df.adj$Open[251:262] - fore.mlp.short$mean)^2)
print(paste("ASE Score:", MLP_SH_ASE))## [1] "ASE Score: 0.24970451814011"c. Neural Network (mlp) - Long horizon
#MLP
NLYLong = df.adj[1:243,]
NLYLongDF = data.frame(Close = ts(NLYLong$Close), High = ts(NLYLong$High), Low = ts(NLYLong$Low))
#Using forecast Open
fit.mlp.Close = mlp(ts(NLYLongDF$Close), hd.auto.type = "cv",reps = 50, comb = "mean")
fit.mlp.High = mlp(ts(NLYLongDF$High), hd.auto.type = "cv",reps = 50, comb = "mean")
fit.mlp.Low = mlp(ts(NLYLongDF$Low), hd.auto.type = "cv",reps = 50, comb = "mean")
#Forecast the explainable features
fore.mlp.Close = forecast(fit.mlp.Close, h = 20)
fore.mlp.High = forecast(fit.mlp.High, h = 20)
fore.mlp.Low = forecast(fit.mlp.Low, h = 20)
#plot(fore.mlp.Open) # plot the forecasts
NLYLongDF_fore = data.frame(Close = ts(fore.mlp.Close$mean), High = ts(fore.mlp.High$mean), Low = ts(fore.mlp.Low$mean))
NLYLongDF_fore## Close High Low
## t+1 10.52282 10.45923 10.32672
## t+2 10.50425 10.44895 10.32348
## t+3 10.50724 10.43844 10.32027
## t+4 10.50509 10.42558 10.31709
## t+5 10.49757 10.41616 10.31395
## t+6 10.49294 10.40806 10.31084
## t+7 10.48778 10.39989 10.30776
## t+8 10.48195 10.39227 10.30471
## t+9 10.47654 10.38526 10.30169
## t+10 10.47112 10.37861 10.29869
## t+11 10.46565 10.37235 10.29572
## t+12 10.46029 10.36648 10.29277
## t+13 10.45498 10.36098 10.28985
## t+14 10.44972 10.35583 10.28695
## t+15 10.44454 10.35101 10.28407
## t+16 10.43943 10.34652 10.28122
## t+17 10.43439 10.34233 10.27839
## t+18 10.42944 10.33843 10.27559
## t+19 10.42456 10.33480 10.27282
## t+20 10.41977 10.33143 10.27007fit.mlp = mlp(ts(NLYLong$Open),reps = 50,comb = "mean",hd.auto.type = "cv",xreg = NLYLongDF) #sensitive to initial values, first 50 iterations
fit.mlp## MLP fit with 7 hidden nodes and 50 repetitions.
## Univariate lags: (1,2,3,4)
## 1 regressor included.
## - Regressor 1 lags: (1)
## Forecast combined using the mean operator.
## MSE: 0.0027.plot(fit.mlp)
NLYDF.Long = data.frame(Close = ts(c(NLYLongDF$Close,NLYLongDF_fore$Close)), High = ts(c(NLYLongDF$High, NLYLongDF_fore$High)),Low = ts(c(NLYLongDF$Low, NLYLongDF_fore$Low)))
fore.mlp.long = forecast(fit.mlp, h = 20, xreg = NLYDF.Long)
plot(fore.mlp.long)
plot(df.adj$Open, type = "l")
lines(seq(243,262,1),fore.mlp.long$mean,col = "blue")
MLP_LH_ASE = mean((df.adj$Open[243:262] - fore.mlp.long$mean)^2)
print(paste("ASE Score:", MLP_LH_ASE))## [1] "ASE Score: 0.114037904415089"c. Neural Network (mlp) - short horizon Environmental Features
#MLP
NLYsmall.eco = finaldata[1:250,]
NLYsmallDF.eco = data.frame(Close = ts(NLYsmall.eco$Close), High = ts(NLYsmall.eco$High), Low = ts(NLYsmall.eco$Low), CPI = ts(NLYsmall.eco$CPI_Score), Morg = ts(NLYsmall.eco$MORTGAGE30US))
#Using forecast Open
fit.mlp.Close.eco = mlp(ts(NLYsmall.eco$Close),reps = 50, comb = "mean")
fit.mlp.High.eco = mlp(ts(NLYsmall.eco$High),reps = 50, comb = "mean")
fit.mlp.Low.eco = mlp(ts(NLYsmall.eco$Low),reps = 50, comb = "mean")
fit.mlp.CPI.eco = mlp(ts(NLYsmall.eco$CPI_Score), reps = 50, comb = "mean")
fit.mlp.Morg.eco = mlp(ts(NLYsmall.eco$MORTGAGE30US), reps = 50, comb = "mean")
#Forecast the explainable features
fore.mlp.Close.eco = forecast(fit.mlp.Close.eco, h = 12)
fore.mlp.High.eco = forecast(fit.mlp.High.eco, h = 12)
fore.mlp.Low.eco = forecast(fit.mlp.Low.eco, h = 12)
fore.mlp.CPI.eco = forecast(fit.mlp.CPI.eco, h = 12)
fore.mlp.Morg.eco = forecast(fit.mlp.Morg.eco, h = 12)
#plot(fore.mlp.Open) # plot the forecasts
NLYsmallDF_fore.eco = data.frame(Close = ts(fore.mlp.Close.eco$mean),
High = ts(fore.mlp.High.eco$mean),
Low = ts(fore.mlp.Low.eco$mean),
CPI =ts(fore.mlp.CPI.eco$mean),
Morg = ts(fore.mlp.Morg.eco$mean))
NLYsmallDF_fore.eco## Close High Low CPI Morg
## t+1 10.50488 10.77812 10.43837 252.4983 4.539445
## t+2 10.52020 10.76670 10.43676 252.5577 4.550443
## t+3 10.50576 10.76732 10.43515 252.6170 4.565469
## t+4 10.50576 10.76966 10.43355 252.8141 4.572542
## t+5 10.51314 10.76791 10.43197 252.8996 4.572679
## t+6 10.51318 10.76645 10.43040 252.9851 4.572563
## t+7 10.51212 10.76565 10.42886 253.0705 4.571883
## t+8 10.51413 10.76451 10.42732 253.2568 4.570938
## t+9 10.51681 10.76323 10.42581 253.3594 4.572390
## t+10 10.51697 10.76198 10.42432 253.4619 4.577478
## t+11 10.51726 10.76068 10.42285 253.5644 4.583795
## t+12 10.51922 10.75930 10.42139 253.7384 4.588483fit.mlp.eco = mlp(ts(NLYsmall.eco$Open),reps = 50,comb = "mean",hd.auto.type = "cv",xreg = NLYsmallDF.eco) #sensitive to initial values, first 50 iterations
fit.mlp.eco## MLP fit with 8 hidden nodes and 50 repetitions.
## Univariate lags: (2,3)
## 4 regressors included.
## - Regressor 1 lags: (1,2,3)
## - Regressor 2 lags: (3)
## - Regressor 3 lags: (2)
## - Regressor 4 lags: (2)
## Forecast combined using the mean operator.
## MSE: 0.0023.plot(fit.mlp.eco)
NLYDF.eco = data.frame(Close = ts(c(NLYsmallDF.eco$Close,NLYsmallDF_fore.eco$Close)),
High = ts(c(NLYsmallDF.eco$High, NLYsmallDF_fore.eco$High)),
Low = ts(c(NLYsmallDF.eco$Low, NLYsmallDF_fore.eco$Low)),
CPI = ts(c(NLYsmallDF.eco$CPI, NLYsmallDF_fore.eco$CPI)),
Morg = ts(c(NLYsmallDF.eco$Morg, NLYsmallDF_fore.eco$Morg)))
fore.mlp.short.eco = forecast(fit.mlp.eco, h = 12, xreg = NLYDF.eco)
plot(fore.mlp.short.eco)
plot(finaldata$Open, type = "l")
lines(seq(250,261,1),fore.mlp.short.eco$mean,col = "blue")
MLP_SH_ASE.eco = mean((finaldata$Open[250:261] - fore.mlp.short.eco$mean)^2)
print(paste("ASE Score:", MLP_SH_ASE.eco))## [1] "ASE Score: 0.229534963042687"c. Neural Network (mlp) - Long horizon Environmental Features
#MLP
NLYLong.eco = finaldata[1:243,]
NLYLongDF.eco = data.frame(Close = ts(NLYLong.eco$Close),
High = ts(NLYLong.eco$High),
Low = ts(NLYLong.eco$Low),
CPI = ts(NLYLong.eco$CPI_Score),
Morg = ts(NLYLong.eco$MORTGAGE30US))
#Using forecast Open
fit.mlp.Close.eco = mlp(ts(NLYLongDF.eco$Close), hd.auto.type = "cv",reps = 50, comb = "mean")
fit.mlp.High.eco = mlp(ts(NLYLongDF.eco$High), hd.auto.type = "cv",reps = 50, comb = "mean")
fit.mlp.Low.eco = mlp(ts(NLYLongDF.eco$Low), hd.auto.type = "cv",reps = 50, comb = "mean")
fit.mlp.CPI.eco = mlp(ts(NLYLongDF.eco$CPI), hd.auto.type = "cv", reps = 50, comb = "mean")
fit.mlp.Morg.eco = mlp(ts(NLYLongDF.eco$Morg), hd.auto.type = "cv", reps = 50, comb = "mean")
#Forecast the explainable features
# Update the forecast horizon to h + forecast horizon
forecast_horizon <- 20
# Forecast the explanatory variables first
fore.mlp.Close.eco <- forecast(fit.mlp.Close.eco, h = forecast_horizon)
fore.mlp.High.eco <- forecast(fit.mlp.High.eco, h = forecast_horizon)
fore.mlp.Low.eco <- forecast(fit.mlp.Low.eco, h = forecast_horizon)
fore.mlp.CPI.eco <- forecast(fit.mlp.CPI.eco, h = forecast_horizon)
fore.mlp.Morg.eco <- forecast(fit.mlp.Morg.eco, h = forecast_horizon)
# Create a data frame with the forecasted values of the explanatory variables
NLYDF.Long.eco_forecasted <- data.frame(
Close = ts(fore.mlp.Close.eco$mean),
High = ts(fore.mlp.High.eco$mean),
Low = ts(fore.mlp.Low.eco$mean),
CPI = ts(fore.mlp.CPI.eco$mean),
Morg = ts(fore.mlp.Morg.eco$mean)
)
# Combine the historical and forecasted explanatory variables
NLYDF.Long.eco_combined <- data.frame(
Close = ts(c(NLYLongDF.eco$Close, NLYDF.Long.eco_forecasted$Close)),
High = ts(c(NLYLongDF.eco$High, NLYDF.Long.eco_forecasted$High)),
Low = ts(c(NLYLongDF.eco$Low, NLYDF.Long.eco_forecasted$Low)),
CPI = ts(c(NLYLongDF.eco$CPI, NLYDF.Long.eco_forecasted$CPI)),
Morg = ts(c(NLYLongDF.eco$Morg, NLYDF.Long.eco_forecasted$Morg))
)
# Fit the MLP model with the combined explanatory variables
fit.mlp.eco <- mlp(ts(NLYLong.eco$Open), reps = 50, comb = "mean", hd.auto.type = "cv", xreg = NLYDF.Long.eco_combined)
fit.mlp.eco## MLP fit with 9 hidden nodes and 50 repetitions.
## Univariate lags: (2,3)
## 4 regressors included.
## - Regressor 1 lags: (1,2,3)
## - Regressor 2 lags: (3)
## - Regressor 3 lags: (2)
## - Regressor 4 lags: (3)
## Forecast combined using the mean operator.
## MSE: 0.0023.plot(fit.mlp.eco)
# Check the length of the updated xreg variable
length_xreg_updated <- nrow(NLYDF.Long.eco_combined)
print(length_xreg_updated)## [1] 263# Proceed with the forecast
fore.mlp.long.eco <- forecast(fit.mlp.eco, h = forecast_horizon, xreg = NLYDF.Long.eco_combined)
fore.mlp.long.eco## Point Forecast
## 244 10.52991
## 245 10.53678
## 246 10.51291
## 247 10.50953
## 248 10.51040
## 249 10.50401
## 250 10.49961
## 251 10.49391
## 252 10.48858
## 253 10.48363
## 254 10.47849
## 255 10.47318
## 256 10.46824
## 257 10.46330
## 258 10.45826
## 259 10.45332
## 260 10.44861
## 261 10.44393
## 262 10.43918
## 263 10.43455plot(fore.mlp.long.eco)
plot(finaldata$Open, type = "l")
lines(seq(242,261,1),fore.mlp.long.eco$mean,col = "blue")
MLP_LH_ASE.eco = mean((finaldata$Open[242:261] - fore.mlp.long.eco$mean)^2)
print(paste("ASE Score:", MLP_LH_ASE.eco))## [1] "ASE Score: 0.10655882823882"d. Ensemble model using at least two of the above. (this model does not have to “beat” your other models.
#Original Data
sh_ensemble = (fore.mlp.short$mean + pred.short$fcst$Open[,1])/2
plot(df.adj$Open, type = "l")
lines(seq(251,262,1),sh_ensemble,col = "green")
lh_ensemble = (fore.mlp.long$mean + pred.long$fcst$Open[,1])/2
plot(df.adj$Open, type = "l")
lines(seq(243,262,1),lh_ensemble,col = "green")
# New Data
sh_ensemble.eco = (fore.mlp.short.eco$mean + pred.short.eco$fcst$Open[,1])/2
plot(df.adj$Open, type = "l")
lines(seq(250,261,1),sh_ensemble.eco,col = "green")
lh_ensemble.eco = (fore.mlp.long.eco$mean + pred.long.eco$fcst$Open[,1])/2
plot(finaldata$Open, type = "l")
lines(seq(242,261,1),lh_ensemble.eco,col = "green")
# Original ASE Scores
ensemble_SH_ASE = mean((ts_nly_open[251:262] - sh_ensemble)^2)
ensemble_SH_ASE## [1] 0.1000075ensemble_LH_ASE = mean((ts_nly_open[243:262] - lh_ensemble)^2)
ensemble_LH_ASE## [1] 0.09386686# New ASE Scores
ensemble_SH_ASE.eco = mean((ts_nly_open.eco[250:261] - sh_ensemble.eco)^2)
ensemble_SH_ASE.eco## [1] 0.1943666ensemble_LH_ASE.eco = mean((ts_nly_open.eco[242:261] - lh_ensemble.eco)^2)
ensemble_LH_ASE.eco## [1] 0.1620551The ensemble does perform better than the Univariate and the MLP models by themselves. It may be worth noting to perform an ensemble model on an ongoing basis.
The new model with the additional economic features vastly under performs in an emesmble approach compared to the original pure stock data.
5. Pick a short and long term forecast horizon based on your “problem” from part 3 and compare all models with the ASE and the rolling window ASE for both the short and long term forecasts … this does not mean you have to choose the model with the lowest ASE.
### Univariate ASEs ###
aic_uni_sh = ase_aic_univariate.short # 0.3818423
aic_uni_lh = ase_aic_univariate.long #
aic_uni_sh.eco = ase_aic_univariate.short #
aic_uni_lh.eco = ase_aic_univariate.long #
bic_uni_sh = ase_bic_univariate.short # [1] 0.3891917
bic_uni_lh = ase_bic_univariate.long #
bic_uni_sh.eco = ase_bic_univariate.short.eco #
bic_uni_lh.eco = ase_bic_univariate.long.eco #
roll_win_ase_short = 0.459
roll_win_ase_long = 0.588
# roll.win.rmse.wge(ts_nly_open, horizon = 12, d = 1 )
# [1] "The Summary Statistics for the Rolling Window RMSE Are:"
# Min. 1st Qu. Median Mean 3rd Qu. Max.
# 0.08954 0.27891 0.40805 0.45852 0.60149 1.49417
# [1] "The Rolling Window RMSE is: 0.459"
### Multivariate Long Short Horizon ###
var_lh = var_ase_long_horizon
var_sh = var_ase_short_horizon
var_lh.eco = var_ase_long_horizon.eco
var_sh.eco = var_ase_short_horizon.eco
mlp_lh = MLP_LH_ASE
mlp_sh = MLP_SH_ASE
mlp_lh.eco = MLP_LH_ASE.eco
mlp_sh.eco = MLP_SH_ASE.eco
comb_sh = ensemble_SH_ASE
comb_lh = ensemble_LH_ASE
comb_sh.eco = ensemble_SH_ASE.eco
comb_lh.eco = ensemble_LH_ASE.eco
# Create a data frame to store the model names and their ASE scores
model_ase_df <- data.frame(Model =
c("AR(7)MA(2) Univariate Short Model", "AR(7)MA(2) Univariate Short Model Eco",
"AR(7)MA(2) Univariate Long Model", "AR(7)MA(2) Univariate Long Model Eco",
"AR(0)MA(0) Univariate Short Model", "AR(0)MA(0) Univariate Short Model Eco",
"AR(0)MA(0) Univariate Long Model", "AR(0)MA(0) Univariate Long Model Eco",
"Rolling Window RMSE Short Model",
"Rolling Window RMSE Long Model",
"VAR Short Horizon", "VAR Short Horizon Eco",
"MLP Short Horizon", "MLP Short Horizon Eco",
"Ensemble Model Short Horizon", "Ensemble Model Short Horizon Eco",
"VAR Long Horizon", "VAR Long Horizon Eco",
"MLP Long Horizon", "MLP Long Horizon Eco",
"Ensemble Model Long Horizon", "Ensemble Model Long Horizon Eco"),
ASE = c(aic_uni_sh, aic_uni_sh.eco,
aic_uni_lh, aic_uni_lh.eco,
bic_uni_sh, bic_uni_sh.eco,
bic_uni_lh, bic_uni_lh.eco,
roll_win_ase_short,
roll_win_ase_long,
var_sh, var_sh.eco,
mlp_sh, mlp_sh.eco,
comb_sh, comb_sh.eco,
var_lh, var_lh.eco,
mlp_lh, mlp_lh.eco,
comb_lh, comb_lh.eco))
# Round the ASE scores to 4 decimal places
model_ase_df$ASE <- round(model_ase_df$ASE, 4)
#Add a new Column "Rank" to the DF which contains there rank order
model_ase_df$Rank <- rank(model_ase_df$ASE)
model_ase_df <- model_ase_df [order(model_ase_df$Rank), ]
# Display the data frame in an output box
print(model_ase_df) ## Model ASE Rank
## 11 VAR Short Horizon 0.0670 1.0
## 8 AR(0)MA(0) Univariate Long Model Eco 0.0859 2.0
## 21 Ensemble Model Long Horizon 0.0939 3.0
## 15 Ensemble Model Short Horizon 0.1000 4.0
## 20 MLP Long Horizon Eco 0.1066 5.0
## 19 MLP Long Horizon 0.1140 6.0
## 17 VAR Long Horizon 0.1551 7.0
## 7 AR(0)MA(0) Univariate Long Model 0.1621 8.5
## 22 Ensemble Model Long Horizon Eco 0.1621 8.5
## 18 VAR Long Horizon Eco 0.1642 10.0
## 3 AR(7)MA(2) Univariate Long Model 0.1671 11.5
## 4 AR(7)MA(2) Univariate Long Model Eco 0.1671 11.5
## 16 Ensemble Model Short Horizon Eco 0.1944 13.0
## 12 VAR Short Horizon Eco 0.2019 14.0
## 14 MLP Short Horizon Eco 0.2295 15.0
## 13 MLP Short Horizon 0.2497 16.0
## 6 AR(0)MA(0) Univariate Short Model Eco 0.3069 17.0
## 1 AR(7)MA(2) Univariate Short Model 0.3818 18.5
## 2 AR(7)MA(2) Univariate Short Model Eco 0.3818 18.5
## 5 AR(0)MA(0) Univariate Short Model 0.3892 20.0
## 9 Rolling Window RMSE Short Model 0.4590 21.0
## 10 Rolling Window RMSE Long Model 0.5880 22.06. Provide the forecasts and prediction limits for both the short and long term forecasts.
# Fit the univariate forecasts AIC & BIC Short Horizions
# AIC Model Forecast
aic_fore.short = fore.arima.wge(ts_nly_open, phi = aic_est$phi, theta = aic_est$theta, d=1,n.ahead = 12, lastn = T, limits = T)## y.arma -0.02 -0.48 0.24 0.17 -0.29 0.21 0.22 -0.04 0.32 0.26 0.14 -0.14 -0.02 0.42 -0.2 0.36 0.09 -0.52 0.09 0.43 -0.19 0.28 0.03 0.05 0.08 0.09 0.02 -0.17 0 0.03 -0.22 -0.37 0.08 0.1 0.08 -0.08 0.19 0.33 0.02 0.12 -0.1 -0.43 -0.04 -0.44 -0.11 0.41 -0.01 0.16 0.04 -0.15 0.16 0.11 0.01 -0.07 -0.03 -0.08 -0.3 -0.06 -0.18 -0.18 -0.07 -0.01 -0.04 0.14 0.08 -0.1 -0.19 0.04 0.27 -0.21 -0.08 -0.06 -0.09 -0.07 -0.13 -0.03 0.08 0.1 0.2 -0.52 -0.01 0.01 -0.67 0.06 0.18 0.28 -0.22 0.4 0.13 0.29 -0.51 0.3 0.03 -0.09 0.27 -0.29 -0.09 0.24 -0.1 0.07 -0.26 -0.48 0.1 -0.02 0.12 -0.19 -0.27 0.31 0.24 -0.44 0.06 -0.41 0.06 0.46 0.22 0.06 0.2 0.23 0.1 -0.01 0.12 0.04 -0.12 -0.17 0.24 -0.17 0.2 0.45 0.15 -0.07 -0.34 0.21 -0.03 -0.2 0.31 0.05 -0.17 -0.14 0.25 0.08 0.11 -0.03 -0.13 -0.13 0.1 -0.49 0.06 0.32 -0.32 -0.42 0.1 -0.04 0.17 0.07 -0.25 -0.07 0.08 -0.13 0.49 -0.25 0.11 -0.31 0.24 -0.11 0.01 0.09 0.06 0.24 0.3 0.23 -0.12 0 0.06 0.12 -0.03 0.19 0.33 0.15 0.06 -0.26 -0.28 0.23 0.28 0.21 0.11 0.26 0.22 -0.56 0.02 0.26 -0.42 0.06 0.04 0.3 0.06 0.01 0.09 -0.07 -0.05 0.12 -0.26 -0.06 0.09 0.02 -0.67 -0.39 0.01 0.29 0.29 0.06 0.12 0.29 -0.2 -0.17 -0.32 -0.26 -0.08 -0.35 -0.56 -0.09 0.38 -0.18 -0.33 0.25 0.27 -0.06 -0.16 0.1 -0.19 -0.03 0.21 -0.13 0.06 0.01 0 0.07 0.05 -0.08 0.16 -0.33 0.24 -0.12 0.07 0.09 -0.03 -0.13 0.27 -0.07 0.04 -0.06 -0.35 0.07 -0.11 -0.11 -0.21 0.07 -0.09 -0.11 0.27 -0.04 -0.04
aic_fore.short.eco = fore.arima.wge(ts_nly_open.eco,
phi = aic_est.eco$phi,
theta = aic_est.eco$theta, d=1,n.ahead = 12, lastn = T, limits = T)## y.arma -0.02 -0.48 0.24 0.17 -0.29 0.21 0.22 -0.04 0.32 0.26 0.14 -0.14 -0.02 0.42 -0.2 0.36 0.09 -0.52 0.09 0.43 -0.19 0.28 0.03 0.05 0.08 0.09 0.02 -0.17 0 0.03 -0.22 -0.37 0.08 0.1 0.08 -0.08 0.19 0.33 0.02 0.12 -0.1 -0.43 -0.04 -0.44 -0.11 0.41 -0.01 0.16 0.04 -0.15 0.16 0.11 0.01 -0.07 -0.03 -0.08 -0.3 -0.06 -0.18 -0.18 -0.07 -0.01 -0.04 0.14 0.08 -0.1 -0.19 0.04 0.27 -0.21 -0.08 -0.06 -0.09 -0.07 -0.13 -0.03 0.08 0.1 0.2 -0.52 -0.01 0.01 -0.67 0.06 0.18 0.28 -0.22 0.4 0.13 0.29 -0.51 0.3 0.03 -0.09 0.27 -0.29 -0.09 0.24 -0.1 0.07 -0.26 -0.48 0.1 -0.02 0.12 -0.19 -0.27 0.31 0.24 -0.44 0.06 -0.41 0.06 0.46 0.22 0.06 0.2 0.23 0.1 -0.01 0.12 0.04 -0.12 -0.17 0.24 -0.17 0.2 0.45 0.15 -0.07 -0.34 0.21 -0.03 -0.2 0.31 0.05 -0.17 -0.14 0.25 0.08 0.11 -0.03 -0.13 -0.13 0.1 -0.49 0.06 0.32 -0.32 -0.42 0.1 -0.04 0.17 0.07 -0.25 -0.07 0.08 -0.13 0.49 -0.25 0.11 -0.31 0.24 -0.11 0.01 0.09 0.06 0.24 0.3 0.23 -0.12 0 0.06 0.12 -0.03 0.19 0.33 0.15 0.06 -0.26 -0.28 0.23 0.28 0.21 0.11 0.26 0.22 -0.56 0.02 0.26 -0.42 0.06 0.04 0.3 0.06 0.01 0.09 -0.07 -0.05 0.12 -0.26 -0.06 0.09 0.02 -0.67 -0.39 0.01 0.29 0.29 0.06 0.12 0.29 -0.2 -0.17 -0.32 -0.26 -0.08 -0.35 -0.56 -0.09 0.38 -0.18 -0.33 0.25 0.27 -0.06 -0.16 0.1 -0.19 -0.03 0.21 -0.13 0.06 0.01 0 0.07 0.05 -0.08 0.16 -0.33 0.24 -0.12 0.07 0.09 -0.03 -0.13 0.27 -0.07 0.04 -0.06 -0.35 0.07 -0.11 -0.11 -0.21 0.07 -0.09 -0.11 0.27 -0.04
# BIC Model FOrecast
bic_fore.short = fore.arima.wge(ts_nly_open, phi = bic_est$phi, theta = bic_est$theta, d=1, n.ahead = 12, lastn = T, limits = T)## y.arma -0.02 -0.48 0.24 0.17 -0.29 0.21 0.22 -0.04 0.32 0.26 0.14 -0.14 -0.02 0.42 -0.2 0.36 0.09 -0.52 0.09 0.43 -0.19 0.28 0.03 0.05 0.08 0.09 0.02 -0.17 0 0.03 -0.22 -0.37 0.08 0.1 0.08 -0.08 0.19 0.33 0.02 0.12 -0.1 -0.43 -0.04 -0.44 -0.11 0.41 -0.01 0.16 0.04 -0.15 0.16 0.11 0.01 -0.07 -0.03 -0.08 -0.3 -0.06 -0.18 -0.18 -0.07 -0.01 -0.04 0.14 0.08 -0.1 -0.19 0.04 0.27 -0.21 -0.08 -0.06 -0.09 -0.07 -0.13 -0.03 0.08 0.1 0.2 -0.52 -0.01 0.01 -0.67 0.06 0.18 0.28 -0.22 0.4 0.13 0.29 -0.51 0.3 0.03 -0.09 0.27 -0.29 -0.09 0.24 -0.1 0.07 -0.26 -0.48 0.1 -0.02 0.12 -0.19 -0.27 0.31 0.24 -0.44 0.06 -0.41 0.06 0.46 0.22 0.06 0.2 0.23 0.1 -0.01 0.12 0.04 -0.12 -0.17 0.24 -0.17 0.2 0.45 0.15 -0.07 -0.34 0.21 -0.03 -0.2 0.31 0.05 -0.17 -0.14 0.25 0.08 0.11 -0.03 -0.13 -0.13 0.1 -0.49 0.06 0.32 -0.32 -0.42 0.1 -0.04 0.17 0.07 -0.25 -0.07 0.08 -0.13 0.49 -0.25 0.11 -0.31 0.24 -0.11 0.01 0.09 0.06 0.24 0.3 0.23 -0.12 0 0.06 0.12 -0.03 0.19 0.33 0.15 0.06 -0.26 -0.28 0.23 0.28 0.21 0.11 0.26 0.22 -0.56 0.02 0.26 -0.42 0.06 0.04 0.3 0.06 0.01 0.09 -0.07 -0.05 0.12 -0.26 -0.06 0.09 0.02 -0.67 -0.39 0.01 0.29 0.29 0.06 0.12 0.29 -0.2 -0.17 -0.32 -0.26 -0.08 -0.35 -0.56 -0.09 0.38 -0.18 -0.33 0.25 0.27 -0.06 -0.16 0.1 -0.19 -0.03 0.21 -0.13 0.06 0.01 0 0.07 0.05 -0.08 0.16 -0.33 0.24 -0.12 0.07 0.09 -0.03 -0.13 0.27 -0.07 0.04 -0.06 -0.35 0.07 -0.11 -0.11 -0.21 0.07 -0.09 -0.11 0.27 -0.04 -0.04
bic_fore.short.eco = fore.arima.wge(ts_nly_open.eco,
phi = bic_est.eco$phi,
theta = bic_est.eco$theta, d=1, n.ahead = 12, lastn = T, limits = T)## y.arma -0.02 -0.48 0.24 0.17 -0.29 0.21 0.22 -0.04 0.32 0.26 0.14 -0.14 -0.02 0.42 -0.2 0.36 0.09 -0.52 0.09 0.43 -0.19 0.28 0.03 0.05 0.08 0.09 0.02 -0.17 0 0.03 -0.22 -0.37 0.08 0.1 0.08 -0.08 0.19 0.33 0.02 0.12 -0.1 -0.43 -0.04 -0.44 -0.11 0.41 -0.01 0.16 0.04 -0.15 0.16 0.11 0.01 -0.07 -0.03 -0.08 -0.3 -0.06 -0.18 -0.18 -0.07 -0.01 -0.04 0.14 0.08 -0.1 -0.19 0.04 0.27 -0.21 -0.08 -0.06 -0.09 -0.07 -0.13 -0.03 0.08 0.1 0.2 -0.52 -0.01 0.01 -0.67 0.06 0.18 0.28 -0.22 0.4 0.13 0.29 -0.51 0.3 0.03 -0.09 0.27 -0.29 -0.09 0.24 -0.1 0.07 -0.26 -0.48 0.1 -0.02 0.12 -0.19 -0.27 0.31 0.24 -0.44 0.06 -0.41 0.06 0.46 0.22 0.06 0.2 0.23 0.1 -0.01 0.12 0.04 -0.12 -0.17 0.24 -0.17 0.2 0.45 0.15 -0.07 -0.34 0.21 -0.03 -0.2 0.31 0.05 -0.17 -0.14 0.25 0.08 0.11 -0.03 -0.13 -0.13 0.1 -0.49 0.06 0.32 -0.32 -0.42 0.1 -0.04 0.17 0.07 -0.25 -0.07 0.08 -0.13 0.49 -0.25 0.11 -0.31 0.24 -0.11 0.01 0.09 0.06 0.24 0.3 0.23 -0.12 0 0.06 0.12 -0.03 0.19 0.33 0.15 0.06 -0.26 -0.28 0.23 0.28 0.21 0.11 0.26 0.22 -0.56 0.02 0.26 -0.42 0.06 0.04 0.3 0.06 0.01 0.09 -0.07 -0.05 0.12 -0.26 -0.06 0.09 0.02 -0.67 -0.39 0.01 0.29 0.29 0.06 0.12 0.29 -0.2 -0.17 -0.32 -0.26 -0.08 -0.35 -0.56 -0.09 0.38 -0.18 -0.33 0.25 0.27 -0.06 -0.16 0.1 -0.19 -0.03 0.21 -0.13 0.06 0.01 0 0.07 0.05 -0.08 0.16 -0.33 0.24 -0.12 0.07 0.09 -0.03 -0.13 0.27 -0.07 0.04 -0.06 -0.35 0.07 -0.11 -0.11 -0.21 0.07 -0.09 -0.11 0.27 -0.04
# Both are more or less the same model given the ARMA component cancel each other out
# Fit the AIC/BIC Picks for Long Horizons
# AIC Model Forecast
aic_fore.long = fore.arima.wge(ts_nly_open, phi = aic_est$phi, theta = aic_est$theta, d=1,n.ahead = 20, lastn = T, limits = T)## y.arma -0.02 -0.48 0.24 0.17 -0.29 0.21 0.22 -0.04 0.32 0.26 0.14 -0.14 -0.02 0.42 -0.2 0.36 0.09 -0.52 0.09 0.43 -0.19 0.28 0.03 0.05 0.08 0.09 0.02 -0.17 0 0.03 -0.22 -0.37 0.08 0.1 0.08 -0.08 0.19 0.33 0.02 0.12 -0.1 -0.43 -0.04 -0.44 -0.11 0.41 -0.01 0.16 0.04 -0.15 0.16 0.11 0.01 -0.07 -0.03 -0.08 -0.3 -0.06 -0.18 -0.18 -0.07 -0.01 -0.04 0.14 0.08 -0.1 -0.19 0.04 0.27 -0.21 -0.08 -0.06 -0.09 -0.07 -0.13 -0.03 0.08 0.1 0.2 -0.52 -0.01 0.01 -0.67 0.06 0.18 0.28 -0.22 0.4 0.13 0.29 -0.51 0.3 0.03 -0.09 0.27 -0.29 -0.09 0.24 -0.1 0.07 -0.26 -0.48 0.1 -0.02 0.12 -0.19 -0.27 0.31 0.24 -0.44 0.06 -0.41 0.06 0.46 0.22 0.06 0.2 0.23 0.1 -0.01 0.12 0.04 -0.12 -0.17 0.24 -0.17 0.2 0.45 0.15 -0.07 -0.34 0.21 -0.03 -0.2 0.31 0.05 -0.17 -0.14 0.25 0.08 0.11 -0.03 -0.13 -0.13 0.1 -0.49 0.06 0.32 -0.32 -0.42 0.1 -0.04 0.17 0.07 -0.25 -0.07 0.08 -0.13 0.49 -0.25 0.11 -0.31 0.24 -0.11 0.01 0.09 0.06 0.24 0.3 0.23 -0.12 0 0.06 0.12 -0.03 0.19 0.33 0.15 0.06 -0.26 -0.28 0.23 0.28 0.21 0.11 0.26 0.22 -0.56 0.02 0.26 -0.42 0.06 0.04 0.3 0.06 0.01 0.09 -0.07 -0.05 0.12 -0.26 -0.06 0.09 0.02 -0.67 -0.39 0.01 0.29 0.29 0.06 0.12 0.29 -0.2 -0.17 -0.32 -0.26 -0.08 -0.35 -0.56 -0.09 0.38 -0.18 -0.33 0.25 0.27 -0.06 -0.16 0.1 -0.19 -0.03 0.21 -0.13 0.06 0.01 0 0.07 0.05 -0.08 0.16 -0.33 0.24 -0.12 0.07 0.09 -0.03 -0.13 0.27 -0.07 0.04 -0.06 -0.35 0.07 -0.11 -0.11 -0.21 0.07 -0.09 -0.11 0.27 -0.04 -0.04
aic_fore.long.eco = fore.arima.wge(ts_nly_open.eco,
phi = aic_est.eco$phi,
theta = aic_est.eco$theta,
d=1,n.ahead = 20, lastn = T, limits = T)## y.arma -0.02 -0.48 0.24 0.17 -0.29 0.21 0.22 -0.04 0.32 0.26 0.14 -0.14 -0.02 0.42 -0.2 0.36 0.09 -0.52 0.09 0.43 -0.19 0.28 0.03 0.05 0.08 0.09 0.02 -0.17 0 0.03 -0.22 -0.37 0.08 0.1 0.08 -0.08 0.19 0.33 0.02 0.12 -0.1 -0.43 -0.04 -0.44 -0.11 0.41 -0.01 0.16 0.04 -0.15 0.16 0.11 0.01 -0.07 -0.03 -0.08 -0.3 -0.06 -0.18 -0.18 -0.07 -0.01 -0.04 0.14 0.08 -0.1 -0.19 0.04 0.27 -0.21 -0.08 -0.06 -0.09 -0.07 -0.13 -0.03 0.08 0.1 0.2 -0.52 -0.01 0.01 -0.67 0.06 0.18 0.28 -0.22 0.4 0.13 0.29 -0.51 0.3 0.03 -0.09 0.27 -0.29 -0.09 0.24 -0.1 0.07 -0.26 -0.48 0.1 -0.02 0.12 -0.19 -0.27 0.31 0.24 -0.44 0.06 -0.41 0.06 0.46 0.22 0.06 0.2 0.23 0.1 -0.01 0.12 0.04 -0.12 -0.17 0.24 -0.17 0.2 0.45 0.15 -0.07 -0.34 0.21 -0.03 -0.2 0.31 0.05 -0.17 -0.14 0.25 0.08 0.11 -0.03 -0.13 -0.13 0.1 -0.49 0.06 0.32 -0.32 -0.42 0.1 -0.04 0.17 0.07 -0.25 -0.07 0.08 -0.13 0.49 -0.25 0.11 -0.31 0.24 -0.11 0.01 0.09 0.06 0.24 0.3 0.23 -0.12 0 0.06 0.12 -0.03 0.19 0.33 0.15 0.06 -0.26 -0.28 0.23 0.28 0.21 0.11 0.26 0.22 -0.56 0.02 0.26 -0.42 0.06 0.04 0.3 0.06 0.01 0.09 -0.07 -0.05 0.12 -0.26 -0.06 0.09 0.02 -0.67 -0.39 0.01 0.29 0.29 0.06 0.12 0.29 -0.2 -0.17 -0.32 -0.26 -0.08 -0.35 -0.56 -0.09 0.38 -0.18 -0.33 0.25 0.27 -0.06 -0.16 0.1 -0.19 -0.03 0.21 -0.13 0.06 0.01 0 0.07 0.05 -0.08 0.16 -0.33 0.24 -0.12 0.07 0.09 -0.03 -0.13 0.27 -0.07 0.04 -0.06 -0.35 0.07 -0.11 -0.11 -0.21 0.07 -0.09 -0.11 0.27 -0.04
# BIC Model FOrecast
bic_fore.long = fore.arima.wge(ts_nly_open, phi = bic_est$phi, theta = bic_est$theta, d=1, n.ahead = 20, lastn = T, limits = T)## y.arma -0.02 -0.48 0.24 0.17 -0.29 0.21 0.22 -0.04 0.32 0.26 0.14 -0.14 -0.02 0.42 -0.2 0.36 0.09 -0.52 0.09 0.43 -0.19 0.28 0.03 0.05 0.08 0.09 0.02 -0.17 0 0.03 -0.22 -0.37 0.08 0.1 0.08 -0.08 0.19 0.33 0.02 0.12 -0.1 -0.43 -0.04 -0.44 -0.11 0.41 -0.01 0.16 0.04 -0.15 0.16 0.11 0.01 -0.07 -0.03 -0.08 -0.3 -0.06 -0.18 -0.18 -0.07 -0.01 -0.04 0.14 0.08 -0.1 -0.19 0.04 0.27 -0.21 -0.08 -0.06 -0.09 -0.07 -0.13 -0.03 0.08 0.1 0.2 -0.52 -0.01 0.01 -0.67 0.06 0.18 0.28 -0.22 0.4 0.13 0.29 -0.51 0.3 0.03 -0.09 0.27 -0.29 -0.09 0.24 -0.1 0.07 -0.26 -0.48 0.1 -0.02 0.12 -0.19 -0.27 0.31 0.24 -0.44 0.06 -0.41 0.06 0.46 0.22 0.06 0.2 0.23 0.1 -0.01 0.12 0.04 -0.12 -0.17 0.24 -0.17 0.2 0.45 0.15 -0.07 -0.34 0.21 -0.03 -0.2 0.31 0.05 -0.17 -0.14 0.25 0.08 0.11 -0.03 -0.13 -0.13 0.1 -0.49 0.06 0.32 -0.32 -0.42 0.1 -0.04 0.17 0.07 -0.25 -0.07 0.08 -0.13 0.49 -0.25 0.11 -0.31 0.24 -0.11 0.01 0.09 0.06 0.24 0.3 0.23 -0.12 0 0.06 0.12 -0.03 0.19 0.33 0.15 0.06 -0.26 -0.28 0.23 0.28 0.21 0.11 0.26 0.22 -0.56 0.02 0.26 -0.42 0.06 0.04 0.3 0.06 0.01 0.09 -0.07 -0.05 0.12 -0.26 -0.06 0.09 0.02 -0.67 -0.39 0.01 0.29 0.29 0.06 0.12 0.29 -0.2 -0.17 -0.32 -0.26 -0.08 -0.35 -0.56 -0.09 0.38 -0.18 -0.33 0.25 0.27 -0.06 -0.16 0.1 -0.19 -0.03 0.21 -0.13 0.06 0.01 0 0.07 0.05 -0.08 0.16 -0.33 0.24 -0.12 0.07 0.09 -0.03 -0.13 0.27 -0.07 0.04 -0.06 -0.35 0.07 -0.11 -0.11 -0.21 0.07 -0.09 -0.11 0.27 -0.04 -0.04
bic_fore.long.eco = fore.arima.wge(ts_nly_open.eco,
phi = bic_est.eco$phi,
theta = bic_est.eco$theta,
d=1, n.ahead = 20, lastn = T, limits = T)## y.arma -0.02 -0.48 0.24 0.17 -0.29 0.21 0.22 -0.04 0.32 0.26 0.14 -0.14 -0.02 0.42 -0.2 0.36 0.09 -0.52 0.09 0.43 -0.19 0.28 0.03 0.05 0.08 0.09 0.02 -0.17 0 0.03 -0.22 -0.37 0.08 0.1 0.08 -0.08 0.19 0.33 0.02 0.12 -0.1 -0.43 -0.04 -0.44 -0.11 0.41 -0.01 0.16 0.04 -0.15 0.16 0.11 0.01 -0.07 -0.03 -0.08 -0.3 -0.06 -0.18 -0.18 -0.07 -0.01 -0.04 0.14 0.08 -0.1 -0.19 0.04 0.27 -0.21 -0.08 -0.06 -0.09 -0.07 -0.13 -0.03 0.08 0.1 0.2 -0.52 -0.01 0.01 -0.67 0.06 0.18 0.28 -0.22 0.4 0.13 0.29 -0.51 0.3 0.03 -0.09 0.27 -0.29 -0.09 0.24 -0.1 0.07 -0.26 -0.48 0.1 -0.02 0.12 -0.19 -0.27 0.31 0.24 -0.44 0.06 -0.41 0.06 0.46 0.22 0.06 0.2 0.23 0.1 -0.01 0.12 0.04 -0.12 -0.17 0.24 -0.17 0.2 0.45 0.15 -0.07 -0.34 0.21 -0.03 -0.2 0.31 0.05 -0.17 -0.14 0.25 0.08 0.11 -0.03 -0.13 -0.13 0.1 -0.49 0.06 0.32 -0.32 -0.42 0.1 -0.04 0.17 0.07 -0.25 -0.07 0.08 -0.13 0.49 -0.25 0.11 -0.31 0.24 -0.11 0.01 0.09 0.06 0.24 0.3 0.23 -0.12 0 0.06 0.12 -0.03 0.19 0.33 0.15 0.06 -0.26 -0.28 0.23 0.28 0.21 0.11 0.26 0.22 -0.56 0.02 0.26 -0.42 0.06 0.04 0.3 0.06 0.01 0.09 -0.07 -0.05 0.12 -0.26 -0.06 0.09 0.02 -0.67 -0.39 0.01 0.29 0.29 0.06 0.12 0.29 -0.2 -0.17 -0.32 -0.26 -0.08 -0.35 -0.56 -0.09 0.38 -0.18 -0.33 0.25 0.27 -0.06 -0.16 0.1 -0.19 -0.03 0.21 -0.13 0.06 0.01 0 0.07 0.05 -0.08 0.16 -0.33 0.24 -0.12 0.07 0.09 -0.03 -0.13 0.27 -0.07 0.04 -0.06 -0.35 0.07 -0.11 -0.11 -0.21 0.07 -0.09 -0.11 0.27 -0.04
# Fit the VAR model using the lagged_data Short
plot(data$Open, type = "l", main = "Forecast of VAR Model Short Horizon Market Open Price at lag 1", cex.main = 0.8)
lines(seq(251,262,1),pred.short$fcst$Open[,1],col = "red")
plot(data.eco$Open, type = "l", main = "Forecast of VAR Eco Model Short Horizon Market Open Price at lag 1", cex.main = 0.8)
lines(seq(250,261,1),pred.short.eco$fcst$Open[,1],col = "green")
plot(data$Open, type = "l", main = "Forecast of VAR Model Long Horizon Market Open Price at lag 1", cex.main = 0.8)
lines(seq(243,262,1),pred.long$fcst$Open[,1],col = "red")
plot(data.eco$Open, type = "l", main = "Forecast of VAR Eco Model Long Horizon Market Open Price at lag 1", cex.main = 0.8)
lines(seq(242,261,1),pred.long.eco$fcst$Open[,1],col = "green")
# Forecast the MLP model Short
plot(df.adj$Open, type = "l", main = "Forecast of MLP Model Short Horizon Market Open Price at lag 1", cex.main = 0.8)
lines(seq(251,262,1),fore.mlp.short$mean,col = "blue")
plot(finaldata$Open, type = "l", main = "Forecast of MLP Eco Model Short Horizon Market Open Price at lag 1", cex.main = 0.8)
lines(seq(250,261,1),fore.mlp.short.eco$mean,col = "green")
plot(df.adj$Open, type = "l", main = "Forecast of MLP Model Long Horizon Market Open Price at lag 1", cex.main = 0.8)
lines(seq(243,262,1),fore.mlp.long$mean,col = "blue")
plot(finaldata$Open, type = "l", main = "Forecast of MLP Eco Model Long Horizon Market Open Price at lag 1", cex.main = 0.8)
lines(seq(242,261,1),fore.mlp.long.eco$mean,col = "green")
# Forecast the Ensemble Models
plot(df.adj$Open, type = "l", main = "Forecast of Ensemble Short Horizon Model for Market Open Price at lag 1", cex.main = 0.8)
lines(seq(251,262,1),sh_ensemble,col = "pink")
plot(finaldata$Open, type = "l", main = "Forecast of Ensemble Eco Short Horizon Model for Market Open Price at lag 1", cex.main = 0.8)
lines(seq(250,261,1),sh_ensemble.eco,col = "green")
plot(df.adj$Open, type = "l", main = "Forecast of Ensemble Long Horizon Model for Market Open Price at lag 1", cex.main = 0.8)
lines(seq(243,262,1),lh_ensemble,col = "pink")
plot(finaldata$Open, type = "l", main = "Forecast of Ensemble Eco Long Horizon Model for Market Open Price at lag 1", cex.main = 0.8)
lines(seq(242,261,1),lh_ensemble.eco,col = "green")
rmarkdown::render(“C:/Users/19405/OneDrive/Documents/MSDS/Time Series/Robert_(Reuven)_Derner_TS_Summer_Project_2023_Final.Rmd”, output_file = “output.html”)
Comment here on the factor, the strong behavior is indicating that
# AIC Model Forecast aic_fore.short = fore.arima.wge(ts_nly_open, phi = aic_est$phi, theta = aic_est$theta, d=1,n.ahead = 12, lastn = T, limits = T)# BIC Model FOrecast bic_fore.short = fore.arima.wge(ts_nly_open, phi = bic_est$phi, theta = bic_est$theta, d=1, n.ahead = 12, lastn = T, limits = T)# Both are more or less the same model given the ARMA component cancel each other out # AIC Model Forecast aic_fore.long = fore.arima.wge(ts_nly_open, phi = aic_est$phi, theta = aic_est$theta, d=1,n.ahead = 20, lastn = T, limits = T)# BIC Model FOrecast bic_fore.long = fore.arima.wge(ts_nly_open, phi = bic_est$phi, theta = bic_est$theta, d=1, n.ahead = 20, lastn = T, limits = T)# AIC Model Forecast aic_fore.short.eco = fore.arima.wge(ts_nly_open.eco, phi = aic_est.eco$phi, theta = aic_est.eco$theta, d=1,n.ahead = 12, lastn = T, limits = T)# BIC Model FOrecast bic_fore.short.eco = fore.arima.wge(ts_nly_open.eco, phi = bic_est.eco$phi, theta = bic_est.eco$theta, d=1, n.ahead = 12, lastn = T, limits = T)# Both are more or less the same model given the ARMA component cancel each other out # AIC Model Forecast aic_fore.long.eco = fore.arima.wge(ts_nly_open.eco, phi = aic_est.eco$phi, theta = aic_est.eco$theta, d=1,n.ahead = 20, lastn = T, limits = T)# BIC Model FOrecast bic_fore.long.eco = fore.arima.wge(ts_nly_open.eco, phi = bic_est.eco$phi, theta = bic_est.eco$theta, d=1, n.ahead = 20, lastn = T, limits = T)ase_aic_univariate.short = mean((ts_nly_open[251:262] - aic_fore.short$f)^2) ase_aic_univariate.long = mean((ts_nly_open[243:262] - aic_fore.long$f)^2) ase_aic_univariate.short #Original = 0.3818423ase_aic_univariate.long #Original = 0.1671257ase_aic_univariate.short.eco = mean((ts_nly_open.eco[250:261] - aic_fore.short.eco$f)^2) ase_aic_univariate.long.eco = mean((ts_nly_open.eco[242:261] - aic_fore.long.eco$f)^2) ase_aic_univariate.short.eco #New = 0.3179122ase_aic_univariate.long.eco #New = 0.08804297ase_bic_univariate.short = mean((ts_nly_open[251:262] - bic_fore.short$f)^2) ase_bic_univariate.long = mean((ts_nly_open[243:262] - bic_fore.long$f)^2) ase_bic_univariate.short # Original ASE [1] 0.3891917ase_bic_univariate.long # Original ASE [1] 0.16209ase_bic_univariate.short.eco = mean((ts_nly_open.eco[250:261] - bic_fore.short.eco$f)^2) ase_bic_univariate.long.eco = mean((ts_nly_open.eco[242:261] - bic_fore.long.eco$f)^2) ase_bic_univariate.short.eco # New ASE [1] 0.3069167ase_bic_univariate.long.eco # New ASE [1] 0.085885roll.win.rmse.wge(ts_nly_open, horizon = 12, d = 1 )roll.win.rmse.wge(ts_nly_open.eco, horizon = 12, d = 1 )# Original Score: # [1] "The Summary Statistics for the Rolling Window RMSE Are:" # Min. 1st Qu. Median Mean 3rd Qu. Max. # 0.08954 0.27891 0.40805 0.45852 0.60149 1.49417 # [1] "The Rolling Window RMSE is: 0.459" # New Score: # [1] "The Summary Statistics for the Rolling Window RMSE Are:" # Min. 1st Qu. Median Mean 3rd Qu. Max. # 0.08954 0.27891 0.40805 0.45852 0.60149 1.49417 # [1] "The Rolling Window RMSE is: 0.459"We can note that there was no difference at all in the RSME from the Rolling window short horizon model with or without the additional economic factors.
roll.win.rmse.wge(ts_nly_open, horizon = 20, d = 1 )roll.win.rmse.wge(ts_nly_open.eco, horizon = 20, d = 1 )# Original Score: # [1] "The Summary Statistics for the Rolling Window RMSE Are:" # Min. 1st Qu. Median Mean 3rd Qu. Max. # 0.1128 0.3796 0.5312 0.5879 0.7586 1.6339 # [1] "The Rolling Window RMSE is: 0.588" # New Score: # [1] "The Summary Statistics for the Rolling Window RMSE Are:" # Min. 1st Qu. Median Mean 3rd Qu. Max. # 0.1128 0.3796 0.5312 0.5879 0.7586 1.6339 # [1] "The Rolling Window RMSE is: 0.588"We can note that there was no difference at all in the RSME from the Rolling window Long horizon model with or without the additional economic factors